I just watched this again. Two things that are guaranteed to happen any time that I watch it are that 1) I'll realize new things about it, and 2) the ending will always affect me the same positive way. I think those are some of the marks of a great movie. Not all of the things I realize are good, though.
I think about 90% of the dialog in the movie was just superb, but McCoy was way over-the-top in that scene where he, Kirk, and Spock are discussing Genesis. It was obvious the writers were just trying to force an argument between he and Spock to be true to the characters, but it wasn't natural. He was barely making a point with his flimsy words there. Thankfully, all of his other lines were natural and/or hilarious. Similarly, I think Shatner was at his most restrained throughout the movie, except when he gave us good ol' hammy Kirk when he was trying to provoke Khan. In that case I didn't mind the theatrics, though, because they made the scene campy in the best and most memorable way.
What I noticed most about the movie this time is how little happens in it. That sounds like a criticism, but I like it for that. It's very economical. If you break down the actual major events, it's not much more than:
1) Saavik takes test
2) Chekov and Terrell go to planet
3) Khan hijacks ship
4) Enterprise goes to space station
5) Khan's ship fights Kirk's ship (twice)
And the rest is mostly just conversations between characters, aside from the eel business. When I realized this, part of me thought, 'that's it?', but another part of me also thought, this is one of those examples of 'less is more'. It's like the complaint I keep making about
"The Dark Knight" (although I like it a lot overall) - so many movies these days seem to cram so much unnecessary story into themselves in order to feel 'epic', it's refreshing to watch a movie that wrings drama more out of characterizations than events.
I didn't find a lot of the major events so exciting or enthralling as conversations between characters, and that happens a lot with good Star Trek episodes and movies. I think two of the biggest reasons why this movie so often wins 'Best Star Trek movie' polls is because the strength of the ending and dialog compel them to forget about or ignore a big flaw in it.
As the years pass and I watch this movie more times, the final battle between the Enterprise and the Reliant in the nebula becomes less and less impressive. This was the first time that, while I watched it, I couldn't stop thinking I was just watching models explode. I can no longer suspend my disbelief. The movie is running so smoothly for such a long time, and then it kinda jumps the rails with some really shoddy special effects in that scene. It's surprising, because I found the rest of the film quite visually impressive. For example, I love that iconic bathed-in-light silhouette hero shot that Kirk is introduced with.
I bet it's easy for a lot of people to overlook that silly battle because it leads to the Spock and Kirk farewell scene, the funeral, and Kirk's conversation with Bones and Dr. Marcus on the bridge, which are a trifecta of awesome.

I keep expecting to watch those three scenes one day and just not be emotionally affected anymore. How could I be when I've seen this movie so many times and own
"Star Trek IV", which means I know Spock will be seen again in another Star Trek movie? It is not logical! And yet that stuff makes me tear up any time.
I think Shatner saying "human", with that hesitation from a lump in his throat, is the finest acting he's ever done. When you add two equally powerful renditions of
"Amazing Grace" to that, I can't stop the waterworks.

While I think the movie stumbles big time with that final battle in the nebula (except for Khan's dialog there, which is a blast to hear), it has maybe the best recovery from a weak scene ever with everything from Kirk running down to engineering until the final shot of the movie.
I like
"Star Trek: First Contact" more overall because there isn't a single moment of that movie I don't enjoy, but I only like this one marginally less because a few rather limp minutes of starships firing at each other don't matter much when everything before and after is so stellar.