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.