Kirk Switches Helmets
Changing Clothes
Music Skip
Spocks Hair
This occurs right after Kirk steps out of the Enterprise into space. When he steps out he is wearing a rectangular helmet. In the next shot he is wearing a round helmet.
At the very end of the movie, Spock and McCoy are on the bridge. Spock's coat has an orange band on the sleeve, and McCoy's has a green band on his sleeve. The shot changes, and all of a sudden, Spock has the green band and McCoy has the orange band.
After Chekov comes back to the bridge after his hand gets burned, there is an obvious skip in the music.
Throughout the movie, The length of Spocks sideburns keep changing.