Two scenes of the Theatrical Version which seem as if they were cut in the Korean Director's Cut due to censorship reasons have probably just been removed because critics argued that Sun-Woo is being hurt a bit to often to be really alive after a while. Therefore, the results of the stabbing on the ice and a few bullet wounds were cut by a tiny bit (not even a second altogether). Of course, I can just assume this has been done for that reason, but it seems highly improbable considering the big amount of violence in the movie that this was cut because of censorship.
The only really important new scene in the Director's Cut explains Sun-woos brutality in Hee-soo's apartment because she tricked him deliberately. The two cut scenes are supposed to prevent the movie from getting too lengthy, especially the first part.
The rest of the differences are small corrections, sometimes with additions, mostly with cuts, which make some things clearer (e.g. the revolver one can see at one of Sun-woo's employees.
The Director's Cut can probably be called the better version but is very common anyway.

16 scenes have been removed, 5 Scenes have been inserted. 2 Scenes were rearranged and slightly lengthened. All in all, the Director's Cut is 30 seconds longer.

Comparison between the German rental DVD (FSK 18) by Splendid and the Korean Director's Cut (Rated 18) by CJ Entertainment.

Removed Scene 2nd minute
The cake Sun-woo is eating in the restaurant can be seen longer.
0,5 seconds

Removed Scene 2nd minute
He chews longer before he takes care of the rowdies.
2,5 seconds

Added Scene 3rd minute
Sun-woo goes to the room with the rowdies with an employee. The man in the background checks his revolver.
2,5 seconds

Removed Scene 10th minute
Sun-woos Boss Kang pulls his hand back after he left the business card on the table.
1,5 seconds

Removed Scene 10th minute
Sun-woo waves at the car for his boss and then opens the door.
8,5 seconds

Removed Scene 15th minute
At the end of the first visit in Hee-soos apartment she notices that Sun-woo forgot his keys. She wants to bring them outside by the run into each other in the corridor. We then see Sun-woo sitting in his car and thinking about the encounter.

Removed Scene 26th minute
Sun-woo watches Hee-soo in the recording studio. When she laughs, he smiles as well. Then one of the technicians throws something over and Sun-woo is startled,
27 seconds

Added Scene 32nd minute
Another man crosses his plans when Sun-woo wants to pick up Hee-woo at the studio again. The Director's Cut now shows the man taking Hee-woo home and the two chatting in front of her house. Then he drives away. Sun-woo drives away as well but is being watched by Hee-woo. When he is gone she calls her friend and tells him that the coast was clear. Sun-woo almost gets into a car accident with the other man who comes driving back because he is calling an employee from the car.
91 seconds

Added Scene 32nd minute
When Sun-woo drives through the tunnel, he recognizes the man he almost crashed into.
6,5 seconds

Added Scene 37th minute
Sun-woo can be seen longer in the car after he drove away from Hee-woo's apartment.
4 seconds

Added Scene 42nd minute
Sun-woo shows some weakness at the end of the encounter with the killer in the parking lot.
4 seconds

Alternative Shots 5th minute
There are different shots being used when Sun-woo meets his boss in the rain. (DC only:) Kang asks him why he kept it a secret and why he did not just call him. He also says that this was not how Sun-woo usually acted. He asks for a reason, but Sun-woo keeps silent.
20 seconds

Removed Scene 86th minute
6 frames of Sun-woo being stabbed in the belly have been removed. Blood is dripping out of the wound.
6 Frames

Removed Scene 87th minute
When Sun-woo kills the man, a short moment in which he can be seen is missing again. The man on the ground is seen shorter as well.
2,5 seconds

Removed Scene 88th minute
Sun-woo can be seen longer in the taxi after he called his boss.
13 seconds

Removed Scene 90th minute
The man whose head was being knocked on the table by Kang is falling down longer.
0,5 seconds

Removed Scene 91st minute
The staff is being driven out of the bar longer.
4 seconds

Alternative Anordung 92nd minute
The scene in which Hee-soo opens the parcel and holds the lamp can be seen after Sun-woo's death in the Theatrical Version. The Director's Cut shows it before Sun-woo enters the bar. Due to continuity reasons, the Director's Cut is longer here.
6 seconds

Removed Scene 94th minute
Sun-woo focuses longer before he shoots his former colleague on the ground.
6,5 seconds

Removed Scene 99th minute
Kang, after being shot in the chest by Sun-woo, stands longer on wobbly legs before he falls down.
4 seconds

Removed Scene 101st minute
One of the gangsters is hiding behind a pillar, breathing heavily.
3,5 seconds

Removed Scene 104th minute
The last enemy behind the table shoots longer at Sun-woo, who is being hit more often.
19 Frames