I found this online.What are the differences in the director’s cut of TWOK?
The release contains a slightly different cut of the film from the theatrical release and the previous DVD release. The run time is slightly longer (approximately 244 seconds for the NTSC version), due to extra footage and different takes being inserted into scenes. Some of the footage comes from extra or alternate footage seen on ABC-TV broadcasts of the film; some of it added specifically for this release, the most notable ones being the two scenes in which it was firmly established that the ill-fated Peter Preston was the nephew of Scotty. At least one line of the extra footage ties directly to a scene from Star Trek IV: The Voyage Home, helping to make the continuity between the movies that much more cohesive.