Both versions contain footage that's missing in the other one, but the Theatrical Version lacks way more footage. Furthermore several scenes have been removed and reinsert afterwards in the TV Version, some of them are edited differently. But these differences aren't mentioned in this comparison because it would break the mold. David Lychn's version is the Theatrical Version btw. He wasn't involved in the TV Version and ordered that his name in the opening credits was going to be replaced by Alan Smithee (common alias for directors who don't want their name in a version). The main difference in the two versions is the fact that the TV Version has been split up to smaller episodes. As a result of that, the movie doesn't look like a motion picture anymore but like a big TV production. Unfortunately, this doesn't make the movie better at all. On the contrary, it makes it worse. Really awful is the aspect ratio because the TV Version comes along in Pan&Scan. A lot of atmosphere gets lost because of this.

Example:
Theatrical Version | TV Version

That's one of the more harmless scenes. There are some scenes where it's way more conspicuous.

Therefore the TV Version is only attractive for people who want anything complete or for hardcore fans of Dune.

Compared are the longer TV Version and the Theatrical Version.
A comparison about the missing footage in the TV Version can be found here.

Running time Theatrical Version: 129:18 min
Running time TV Version: 176:47 min

Missing footage Theatrical Version: 3073 min = 51:12 min
The difference in running time is due to roundings of the removed footage, short black images (ad) in the TV Version and probably some scenes I missed (sorry if that is really an issue).

01:40 Min
Different prolog in either of the versions. The camera slowly tracks to Frank Herbert's Dune, followed by a narration about the Dune universe, the planets, the lords and the events. The whole thing is being displayed in drawings and accompanied by a narrator. The end of the prolog contains an extended shot of a landing plane.
458 sec

05:05 Min
The emperor asks hisself why guild had sent a third-grade navigator. He concludes that it had to be about the spice. Guards arrive in the background, they report that Reverend Mother Gaius Helen Mohiam had been chosen and ask if they were dismissed. The emperor sets them free and wishes not to be interrupted anymore. Then an extended shot of Reverend Mother Gaius Helen Mohiam walking in the corridor and finally the emperoro for s ahort period. Furthermore there's an additional sign in the middle of the screen which reads "Arrival of the navigator".
42 sec

05:17 Min
A shot of Helen Mohiam during the end of the sentence "...Report when we finish" in the TV Version.
(2 sec) no difference
TV Version on the left, Theatrical Version on the right

05:18 Min
Extended conversation between the emperor and Helen Mohiam. She explains spice causes changes in evolution and she wanted to be close to the emperor.
24 sec

07:00 Min
Shot of Helen Mohiam using telepathy during the question "How was your journey?" in the TV Version.
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

07:26 Min
Missing part of the dialog between the emperor and the navigator. The emperor wants to know if there was a problem and adds that there usually was when the navigator paid him a visit. The navigator responds the answer was in the problem.
10 sec

07:40 Min
Different perspectives of the navigator's dialog about its being aware of the hostility of the two houses and its suspicion that the emperor knows more. The Theatrical Version contains footage of the emperor from behind, the TV Version shows the emperor's front.
(10 sec) no difference
TV Version on the left, Theatrical Version on the right

07:56 Min
Shot of the navigator during the dialog. In the Theatrical Version, it's the emperor instead.
(2 sec) no difference
TV Version on the left, Theatrical Version on the right

08:02 Min
Shot of Helen Mohiam doing telepathy. It's the emperor and the navigator in the Theatrical Version.
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

08:26 Min
Extended dialog of the emperor and the navigator. The navigator gets angry and tells him that spice had given him lifetime of 4000 years and that he was going to live only 200 more years. He gives the metaphysical skill to the Bene Gesserit Sisters. The emperor was the highest authority. The entire power of the universe is in spice. If the empire gets lost, planets get lost as well and so soes the emperor.
36 sec

08:47 Min
Different shot of thw navigator during the dialog.
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

08:53 Min
The shot at the end of the dialog is from the emperor's POV in the Theatrical Version. In the TV Version, it's from the other side.
(5 sec) no difference
TV Version on the left, Theatrical Version on the right

09:26 Min
Dialog between the emperor and Helen Mohiam. She says the Atreids on Arrakis were supposed to split the spice production. She only needed a slim amount to extend her life. But the Atreids needed a tremdous amount of spice. Then she says she was worried and adds anybody was worried and releases him. The arriving guards are being informed that she was a Helen Mohiam and she can't be influenced by anyone in the entire universe. She was more loyal to the sisterhood than to the emperor. The guards should take good care of her. At the end, he asks hiself why they wanted the duke's son dead.
63 sec

09:43 Min
Helen Mohiam on the way to Arrakis with the ship. Meanwhile, she thinks about the last conversation. Intercuts to the two pilots and the exterior view of the ship.
47 sec

11:19 Min
The Theatrical Version contains a shot of ring coming closer to the camera. The TV Version instead contains some shots of the planet.
(5 sec) no difference
TV Version on top, Theatrical Version below

11:36 Min
Two of the three men coming in are being introduced by an announcer. Guerney Halleck, friend and teacher of Paul Atreides, and Dr. Wellington Yueh, royal physican of Atreid.
27 sec

11:43 Min
Now the third character is being introduced by the narrator: Thufir Hawat.
20 sec

14:22 Min
Extended conversation between Paul and Dr. Wellington Yeuh. Paul wants to know if it was true that the sand moves with 700 km/h. Dr. Wellington Yeuh is affirmative and comes up with more info about the storms.
27 sec

17:12 Min
Extended shot of Paul fighting the machine. Intercuts to Guerney Hallek and Thufir Hawat. Hallek says it wasn't his or Duncan's influence but his mother's.
23 sec

18:10 Min
Shot of the sea. The camera pans up the tower. Paul's father is standing on the balcony - another shot of the sea.
21 sec

18:23 Min
Missing shot of Paul. He says moving into a new house and leaving the old baggage behind was good. Then a shot of Paul again.
9 sec

21:30 Min
Extended conversation between Paul's mother and Helen Mohiam (Bene Gesserit). They're talking about Paul. His mother is worried because he is on Arrakis. At the end of the conversation, Helen Mohiam (Bene Gesserit) tells Paul was going to be tested in a couple of minutes because they needed to find out if he was a living human being or a dead animum. Having finished that sentence, Helen Mohiam (Bene Gesserit) can feel Paul's presence. There's a sign that reads "Jessican and Reverend Mother Gaius Helen Mohiam" being displayed in the middle of the screen.
82 sec

22:50 Min
Shot of Paul. His mother turns around one more time and says to him he shouls remember he was the duke's son.
7 sec

23:21 Min
Shot of Helen Mohiam.
2 sec

24:57 Min
Shot from the bottom up when Paul puts his hand in the box. In the Theatrical Version, the angle is different and 3 sec longer.
3 sec
TV Version on top, Theatrical Version below

25:03 Min
Helen Mohiam's dialog that he shouldn't move his hand when he can feel the poison.
11 sec

27:08 Min
Right before the end of the sentence "Maybe. It will be houres to controll": shot of Paul in the TV Version, shot of Helen Mohiam in the Theatrical Version.
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

27:11 Min
Right after the alternate angle, a short dialog scene with Helen Mohiam, who asks whether or not he recognized anyone telling the truth, follows. He replies he could recognize it.
8 sec

27:15 Min
The previous scene follows a different scene here. Helen Mohiam puts his hand on Paul's forehead and removes it a couple of seconds later. The hand removal is shown from a completely different angle and comes presumably from the cut before. A dialog about Paul's dreams follows. She mentions Paul might be Kwisatz Haderach. At the beginning of the dialog, "the exam" is being displayed. The sentence "Dou you now..." from the Theatrical Version is missing here. Instead the TV Version comes up with "of whater of live", not "Do you now of whater of live" (Theatrical Version).
30 sec

28:36 Min
Shot of sad Paul.
2 sec

28:43 Min
Helen Mohiam turns around to Paul and says the voice didn't protect him. Subsequently the (slightly different) shot of Paul. Then a shot of Helen Mohiam and Paul's mother leaving the room. After that scene, a shot of the planet itself follows.
64 sec

29:07 Min
Shot of the ring imprint.
3 sec

33:53 Min
Shot of one of the baron's helper.
1 sec

34:08 Min
When the TV Version ends, a shot of a planet follows. Then an exterior shot of the buildung, followed by a shot of Paul's parents cuddling in bed, bandying words. At the beginning of the scene, "Alias' procreation" is being displayed.
79 sec

34:54 Min
Shot of a building.
2 sec

35:53 Min
Exterior shot of the ship plus additional shot of Paul and his parents.
2 sec

37:22 Min
The "navigator at work" scene has been entirely removed. The navigator flies/swims (tough to recognize) in space and fires off some rays with the body. Due to those rays, new planets develop.
94 sec

38:31 Min
"The people of Fremen". Fremen passes a mountain to get to the landed space ship. Meanwhile the people of Fremen is being explained via voice-over. Then a couple of soldiers on their way to the space ship with a flag. Cut to one of the Fremes with binoculars in his hands.
51 sec

38:38 Min
Another shot of the Fremes with the binoculars. He puts it away and says "The mother and the son".
6 sec

39:16 Min
Duncan delivers a message from the Fremes to Paul's father. He reads it out loud and talks to Duncan in between.
33 sec

40:52 Min
Now footage with a running time of approx. 16.5 min in the TV Version without interruptions, which follows slightly later in the Theatrical Version.

- The end of the scene with Yueh getting reading out the message from the body is missing. (43 sec)

- Paul passes a corridor. Cut to a room with Paul, he is sitting there with a board in his hands, "Paul takes spice" is being displayed. A short explaination of the spice transporters follows. After that, Paul walks up to a box of spice on the table. (41 sec)

- "Lelo's plans for the future". Paul's father asks Thufir about a radio device if they had found sth. Thufir denies. Paul's father is worried about the traitor plus shots of the soldiers. (21 sec)

- The beginning of the discussion has been shortened. (42 sec)

- So has the end. (118 sec)

- "The prophecy". Bene Geserit Mother goes to the concubine. Both shortly talk to each other and it turns out that she's the one she's been waiting for. (68 sec)

333 sec

43:27 Min
Shot of the soldiers plus getting in the space ship. Furthermore another shot of the soldiers plus the ship taking off.
22 sec

46:13 Min.
An additional exterior view plus a short dialogue.
17 sec.

56:50 Min.
Short exterior view of the palace. Then follows the chapter "The Encounter". Gurney plays a song for the attendees. Then follows a short discussion. Over the course of this discussion about the spy, Paul's father spills a cup of water on the ground. Then there's a shot of a spaceship. After that scene Paul sits in a room and looks at the imprint of the ring - a voice from off-screen tells you a little bit about the ring. Then, Paul's father enters the room and talks to his son. When a guard arrives and says that there's something on the screen, he leaves his son again.
235 sec.

60:34 Min.
Exterior view of several space ships. Then you see the baron and assistant sitting at the wheel. Then the movie cuts back to the station. Thufir looks into one of the rooms and sees one of the modules burning (7 seconds of Thufir seeing the burning wall were used in the Theatrical Version at 1:01:13). Then you see another shot of the space ships + the baron and his assistant.
33 sec.

60:54 Min.
Again, there's a shot of the space ships + the baron and his assistant.
6 sec.

61:01 Min.
A space ship lands.
5 sec.

61:34 Min.
Several soldiers leave the station. Then you see the soldiers walking along a hallway.
7 sec.

61:43 Min.
A few shots of the fortress being attacked.
5 sec.

3 short scenes during the next few seconds were mirror-inverted for the TV-Version.

62:17 Min.
You see some more war scenes.
24 sec.

62:54 Min.
"The Emblem". Yueh is led into a room by the guards. Feyd puts his hands over Paul's fathers nose and mouth so that the latter almost suffocates. Yueh apprehensively watches these events. After the choking, Sting puts the poison (which Paul's father had in his mouth) in Yueh's mouth. Then the baron looks at Paul and talks to Jessica. 11 seconds of the choking were also used in the Theatrical Version.
99 sec.

63:31 Min.
Short shot of Jessica's face.
3 sec.

64:15 Min.
A guard enters a room to visit Piter. Then follows a short shot of Jessica's face.
12 sec.

64:30 Min.
A few war scenes. Then you see Piter going to Jessica to talk to her. Shortly afterwards, a few gaurds carry Jessica and Paul on a stretcher. In between, Paul shortly opens his eyes.
100 sec.

65:13 Min.
Yueh is led to the baron by the guards. The baron is scared and the traitor as well as the guards leave the room.
19 sec.

65:52 Min.
The stabbed Yueh is carried out of the room. Subsequently a soldier enters and Piter asks him where they are. The soldier replies that they are in the great desert.
29 sec.

66:00 Min.
Short shot of Paul, Jessica, and the pilot.
3 sec.

66:48 Min.
Short shot of Jessica.
2 sec.

68:36 Min.
"The Baron, Piter and Leto". Soldiers carry Paul's father to the baron.
19 sec.

68:47 Min.
Now follows a short conversation between the baron and Piter.
32 sec.

70:50 Min.
A short panning shot from Paul to Jessica.
7 sec.

71:17 Min.
It takes a little longer for them to get out.
2 sec.

The image was brightened up.

80:02 Min.
You can see a "Plumbser".
3 sec.

83:03 Min.
There's an overlay that says "Paul fights with a Fremen". A Fremen comes down from the stones. He challenges Paul to fight with him. The two of them search for some free space and then start to fight - they both use a knife. At the end of the fight, Paul wins - the Fremen is dead.
119 sec.

83:57 Min.
You see the Reverend Mother from the beginning of the movie talking about the prophecy. Then follows the scene of the funeral ceremony. The murdered Fremen is lying on a stretcher in front of an incinerator. His wife and son approach the dead man and place a little object next to him. Then they push him in the incinerator. 2 other Fremen pour some water into a jar. Then the Fremen's leader gives a necklace to Paul. After the whole procedure everyone leaves the room.
164 sec.

85:26 Min.
"The Death of Chai's Father". Paul's father and his wife Jessica go to Paul. The father tells Paul that Chani's father is dead. While he says that the movie cuts to a scene where the baron kills him. After the conversation a group of Fremen pass Paul. At the end there are two short shots of the exterior of the palace (both had already been used before).
50 sec.

89:10 min
During the birth of the baby, a short scene showing the Reverend Mother screaming. The Theatrical Version only shows the baby.
3 sec

89:18 min
Shot of a planet followed by the exterior view of the space base in which a wagon drives in. Then we can see a higher view of the hangars with one of the wagons entering the picture. Eventually, we see the wagon stopping and the guards saluting.
26 sec

95:22 min
Paul passes the soldiers and approaches on of the men with Kiswa hook. When he arrives, there is an overlay stating that Stilgar was explaining the rider ritual and the two talk shortly. This seems to be another take as the hook is in another angle, Paul moves his hand differently when he takes the hook and a few hair of someone in the background are tilted in another diretion.
30 sec

95:33 min
Short dialog in which the man explains to Paul how he has to proceed. At the end, he is putting a [] on his shoulders.
56 Sek

99:16 min
A few men are running out of the Spice harvester. The next scene also features an overlay declaring it a “longer war scene“.

4 sec

99:26 min
Soldiers are running down a sand hill. One of the warriors screams something and shoots. Soldiers are being hit and fly through the air. Paul and his colleague are watching the happenings. Soldiers are running through the picture. The friendly soldiers are shooting at just those running ones.
24 sec

99:35 min
A soldier is flying through the air + short view of the shooting soldiers + some more war scenes.
13 sec

101:47 min
"Origin of the Water of Life“. The man who has accompanied Paul during the attack now shows him the origin of the water of life. In a big funnel full of water there is a sandworm sucking in the water. Then two are taking the worm out of the funnel and strangle mucus out of it. This blue mucus is being poured into a container. Paul looks ihm heraus. Dieser blaue Schleim
wird danach aus einem Hahn in ein gefußt gegossen. Paul looks amazed.
90 sec

109:20 min
The lord can be seen a bit longer + exterior view of the fortress.
9 sec

109:48 min
Tracking shot towards the fortress + another view of the space glider.
11 sec

109:54 min
The crew of the glider can be seen.
25 sec

110:28 min
Exterior shot of the fortress + shot of the lord.
6 sec

110:34 min
Paul, his aid and Gurney go down the stairs to the control room.
28 sec

112:12 min
The baron is flying a bit further towards the emperor. Then the rest of the flight can be seen from the baron's view and the head lying on the ground.
5 sec

112:42 min
The emperor and the baron are talking longer.
17 sec

113:41 min
Short view of Paul + exterior view of the fortress.
5 sec

114:09 min
View of Paul, his aid and Gurney in their protective suits.
3 sec

114:27 min
Paul takes his suit off.
2 sec

114:34 min
Paul goes away.
3 sec

115:43 min
Soldiers are running out of the fortress.
6 sec

115:47 min
Shot of a sand worm + soldiers climbing up a dune.
18 sec

116:30 min
A space ship is being blown up.
2 sec

119:33 min
A few soldiers are fleeing into the fortress.
3 sec

124:51 min
The TV version shows us the desert and clouds gathering for a storm instead of the sea.
10 sec

125:01 min
Contracting clouds and the following lightning can be seen.
5 sec

127:22 min
The TV version, there are more ending credits: “Prologue Graphics Concept by FRANCESCA TURNES and Prologue Paintings by JAROSLAV GEBR“
5 sec