The extended argument between McCoy and Kirk that was in the SLV, was one of my favorite bits. It was both funny ("And another thing..." "Get out of here, Bones!") and badass ("command fitness"). I always wondered why it didn't make it into the DE. Especially since it contains absolutely no special effects, and the theatrical/DE version of the scene is so badly edited - De Kelley's lines are obviously looped.
Another thing I like about the DE is that it actually shows San Francisco. The theatrical version (in the one or two seconds that we actually see this - it's when we see the Golden Gate Bridge) shows bare land & forest where the city is supposed to be. This is because Gene Roddenberry had an appalling opinion that all cities on Earth would be underground!

I'm so glad the DE got rid of that nonsense.
I also like the fact that the DE did things like this:
- got rid of that horribly grating Epsilon 9 computer voiceover when the Klingon ship's log is being played back. (I never realized until now that the computer is actually giving an English translation
of the lines spoken by the Klingon captain in that log.)
- slightly altered the English translation of the words spoken by the Vulcan masters at Spock's Kolinahr ceremony. In the theatrical version it is completely obvious that the Vulcan words were created after the fact, to match the lip movements of the actors, who were speaking English during the scene. This is also true of the conversation between Spock and Saavik in ST II, but that time, it wasn't so bloody obvious - no "gib me yu troats!"
- gives a MUCH better depiction of Vulcan itself. For example, we learn that the rock formations right behind the Kolinahr masters are actually the legs of a huge statue right behind them. And it gets rid of the not-moon...
One nitpick about the DE: As Admiral Kirk's air tram is approaching Starfleet Command, the DE shows the scene from the outside, where we see that the shuttle is heading for a point that looks to be hundreds of feet up from the water. But the inside of the building (which is all we see in the theatrical version) is level with the water's edge...