I think it has the best overall character development in all the Treks. I think it's no surprise with Nimoy directing that he would know how to handle these characters, all the more disappointing when Shatner couldn't do the same in Trek V.
The worst I can say about it is that the cinematography has a decidedly flat TV look. Trek II sometimes did as well, but not quite so much. The Genesis Planet in particular looks almost TOS-like sometimes for being on a soundstage. Imagine if they had just gone on location to a real jungle instead. Only a few shots look epic on Genesis, like the fiery end of the Enterprise on the outcropping and Kruge's death (presumably courtesy of ILM).
It hadn't been many years since Christopher Lloyd was on Taxi so it was really, IMHO, a casting mistake to have him as the heavy. His character on Taxi was most well-known for his voice and no amount of makeup or speaking his lines in Klingon could completely erase my image of him being a comedic actor from Taxi. Ricardo Montalban certainly did a better job shaking his then-current image of Mr. Roarke in Khan than Lloyd did with Kruge. With the passage of time, and Lloyd playing other heavies in things like Roger Rabbit, he seems more believable.
I like Robin Curtis' screen presence, but she didn't seem particularly vulcan to me. She seemed too soft and sympathetic, so the recast hurt the picture as well.
I think Horner's score for III is better than II, definitely the emotional backbone. II goes over the top too often, almost to the point of camp with some of the Khan themes, and III is a more mature work.
It's really III that marked the transition of the cast to more of an ensemble. That's not how it may have been conceived in the 60s, but it's how they evolved, through the influence of their continued popularity with fandom. I found that very satisfying vs, let's say, TMP where the crew don't seem to have the same familial bond.