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).

00:00 Min
Missing "Constantin Film" Logo, followed by a black screen.
7 sec

00:17 Min
After the collimation "A David Alan Smythe" (it's just David Lynch in the Theatrical Version), a woman comes up with some deets about spice and the planet Arakis (Dune).
97 sec

01:40 Min
The following prolog is completely different in either of the versions.

Theatrical Version:
Classified report about the conspiracy to sabotage the spice extraction on the screen. Four planets (Arrakis, Caladan, Giedi Prime and Kaitain) are under suspicion and a third-degree navigator ought to be send to Kaitain to get information about the conspiracy from the emperor. "The spice needs to flow"

39 sec

10:10 Min
The emperor turns around and drops his coat. Then a high-angle shot of the space harbor.
9 sec

11:32 Min
During the sentence "...Report when we finish", the TV Version contains a 2 sec-shot of Helen Mohiam (Bene Gesserit).
(2 sec) no difference
TV Version on the left, Theatrical Version on the right

13:03 Min
Shot of the navigator's mouth (?) in the Theatrical Version. The TV Version contains a shot of the emperor in the first 2 sec instead.
(4 sec) no difference
TV Version on the left, Theatrical Version on the right

13:03 Min
The TV Version contains a shot of Helen Mohiam at the telepathy during the question "How was your yourney?".
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

13:51 Min
After the missing dialog in the Theatrical Version:
Alternate angle when the navigator explains it was aware of the hostility of the two houses and adds the emperor was hiding sth. The Theatrical Version shows a third person perspective from the emperor most of the time, the TV Version shows the emperor's front instead.
10 Sek.sec
TV Version on the left, Theatrical Version on the right

14:22 Min
Shot of the navigator during the dialog - shot of the emperor in the Theatrical Version.
(2 sec) no difference
TV Version on the left, Theatrical Version on the right

14:28 Min
Helen Mohiam is transmitting thoughts again - shot of the emperor and an extended shot of the navigator in the Theatrical Version.
(3 sec) no difference
TV Version on the left, Theatrical Version on the right

15:47 Min
Shot from the emperor's POV at the end of the dialog in the Theatrical Version, shot from the other side in the TV Version.
(5 sec) no difference
TV Version on the left, Theatrical Version on the right

20:05 Min
Shot of the ring coming closer to the camera in the Theatrical Version. The TV Version contains a couple of shots of the planet instead.
(5 sec) no difference
TV Version on top, Theatrical Version below

38:40 Min
Here's one more dialog scene, but therefore another scene is missing. The Theatrical Version contains a shot of a Freme asking Paul to tell about about his homeland, followed by a shot of Helen Mohiam's face. She puts her hand off Paul's forehead and says "Do you now..."
6 sec

43:39 Min
Shot of two of the baron's assistants plus shot of blood flowing in a jar plus a slightly extended tracking shot to the baron's face.
9 sec

During the following conversation with the baron, the voices are deeper and anything is a little slower (that just sucks!).

45:41 Min
Shot of one the baron's assistants, then a removed shot of the room. Close-up of one the two guys who just came in, he has a worm or sth. in a jar. He pushes the cover and squashes the worm, then he drinks sth. through a straw. The baron laughs. While the baron's assistant is kissing ass, a man is being brought in by two security guards. He puts some flowers in a couple of bins. While doing, the baron says "I will have Arakis back for myself!". The TV Version shows a shot of two guys during that sentence instead.
46 sec

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47:29 Min
The baron's bloodbath is longer. He goes to the man, who was brought in by the security guards, and pulls the plug out of his heart. Blood is leaking and the baron pushes him against the wall. Then a couple of shots of the present people. Finally blood splatters on the wall and the man slides down the wall, leant against the baron.
46 sec

49:17 Min
After Paul looked at his mother, a shot of her is missing. She thinks at the night with her husband. While being in bed, they hug and she tells him she was going to miss Caladan.
21 sec

74:24 Min
Two soldiers at the radio device, they're listening to chitchatting Harkonns.
22 sec

85:51 Min 03:30 4. Chapter (43:27)
The camera pans to the door, then an extended shot of the Bene Geserit Mother's shadow in a corridor.
10 sec

3 short scenes in the following seconds are mirrored in the TV Version.

92:08 Min
Shot of weeping Yueh.
13 sec

94:02 Min
The baron also tells Jessica he'd love to spit on her head before leaving, then he spits in her face.
12 sec

94:25 Min
One of the baron's assistants is mocking Jessica.
22 sec

99:22 Min
Shot of the entire crew - both pilots, Jessica and Paul.
3 sec

101:52 Min
Piter bends over Paul's father, then he looks at the baron.
6 sec

111:09 Min
Paul explains he was going to take a dump to confuse the worm. Jessica goes down slowly. The image of the Theatrical Version slightly judders, the TV Version lacks this footage.
12 sec

111:11 Min
Jessica watches Paul. When he's done, he goes to Jessica and reminds her of not walking rhythmic because they didn't want to provoke the worm.
23 sec

111:17 Min
Paul and Jessica are walking up a dune, followed by two intercuts.
12 sec

116:45 Min
The men leave.
6 sec

116:48 Min
Shot of a stranger. He kicks backwards and another Freme is standing there.
4 sec

115:55 Min
Shot of the baron saying "Fade" in the Theatrical Version. The TV Version lacks this shot, the baron says "Fade" offscreen.
1 sec

118:00 Min
A part of the birth is missing. In the Theatrical Version, Reverend Mother shortly screams.
3 sec

118:08 Min
Shot of sth. what can't really be recognized. It looks loke a space station or sth. Then a shot of a hose while the remaining meat is being removed from the ground.
7 sec

131:55 Min
The woman fires off another shot and hits the sand. As a result of that, some soldiers are being tossed around.
2 sec

141:09 Min
Shot of Alia with her mother. She feels that Paul had drunken some of the water of life. Helen Mohiam dies.
26 sec

148:03 Min
Shot of the emperor yelling angrily.
2 sec

154:04 Min
Shot of the firing soldiers.
1 sec

203:53 Min 04:32 7. Chapter (2:04:51)
Shot of the sea instead of the desert and the clouds before a storm.
4 sec

Missing in the TV Version:
04:32 7. Chapter (right at the end)
PG-13 logo + "Deutsche Bearbeitung Cina - Adaption Renate Wolf" collimation
36 sec

The final credits, when some of the actors are being introduced with a picture, are 10 sec shorter in the TV Version.