Let's get this one out of the way right now.
1: Star Trek II
2: Star Trek I
Was there ever going to be any real doubt of that outcome? Really? It's like asking 'which is better, Phantom Menace or Empire Strikes Back?' (To be fair, if anyone wants to defend that, by all means write a comment.) Usually received fan wisdom is next-door to worthless, but in this particular case, it's pretty damn spot on. Wrath of Khan isn't just a good Star Trek movie, it holds up well simply as a movie in its own right, and that's not a common thing in science-fiction franchises.
The plot is compelling, fast-paced, and concentrated on the stories of the characters rather than tied into amazing special effects. This time, the focus is fully on the original cast – who let's face it, are what the audience still turned up to see, though with Ricardo Montalban at his bombastic best to serve as the bad guy. What we have here is truly a character-piece, boiled down to one element – everyone is growing old. The characters are now in their forties and fifties, and Kirk especially is feeling it. Then he meets the son that circumstances have pretty much kept in from, and seems to get the option of rekindling a relationship with the mother...while meanwhile a big loose end from the past is seeking to kill him. Best of all – a reference to the original series that isn't a retcon, isn't just a pathetic copy of the original plot in a desperate attempt to wave a flag to the fanboys.
Cast first. This time it's a Shatner-focused story, where arguably last time it was...well, all over the damn place. He's got to carry the whole weight of the plot on his shoulders, but it fits comfortably and well. Conjuring up chemistry with the Marcuses isn't exactly easy given everything else going on around him, but especially with Carol Marcus it fits extremely well – you can quite easily believe that they were once involved. With David it's a bit harder, but for one thing he has a lot less time on screen with him, for another the actor involved is a bit, well, wooden. When it comes to the angst of aging, again, he manages to portray that well without exagerration. Bluntly – if it wasn't for the next one, I'd label this his best-ever performance.
Nimoy, less to do this time except die. The thing that attracted him to the movie in the first place, the opportunity to manage a great death scene. Watching this again, he really is 'on second' this time, providing a supporting role for Kirk until he gets to his big payout at the end of the movie – and that is one of the biggest scenes ever in Star Trek, the hammerblow of watching Spock die and then transitioning to his funeral. Kelley, solid as ever in the supporting role, this time basically hanging around with Kirk and trying to snap him out of his funk. As for the others – well, Koenig gets a lot more to do in this movie than in the previous one (or any of the other films), promoted to First Officer and given the job of getting the plot kicked off, as well as betraying Kirk – for a while. The man can scream most effectively.
Then Montalban, who is a good match for Kirk – in that both have the same capability to rage at the screen, chewing the scenery to the extent that you can clearly see the teeth marks in the bridge set. He's a very one-dimensional character, focused entirely on his desire for revenge. That is potentially a little weakness here, as it makes him rather one-note; despite one of his people pointing out that he has a universe-conquering tool down in his transporter room, he instead decides to tick 'kill Admiral Kirk' off his bucket list. Let's just hope that was one of the last items on that list.
The movie looks right. I could argue that the first movie looks more 'realistic', but the second movie looks like the Hornblower in Space that we've all related Star Trek too in the past; Joe Jennings talks in one of the on-disc documentaries about how the director wanted to 'run out the guns' and so forth – it looks right, and it works well. Perhaps it leans a bit too heavily towards the 'military' side – but put it this way, they don't bother changing the uniforms again for the rest of the franchise.
Someone's watched the movie and the series here, and worked out what works and what doesn't. That knowledge benefits the movie enormously – and that it is still the fan favourite today is testament to that.