Not a little detail so much as a broad one that may have been very obvious to others all along...but last night I put on TWOK in commemoration of Leonard Nimoy's passing, and realized just how much it's Kirk's film...Spock dies in it, but even that is played for how it affects Kirk. It wouldn't be too much of an exaggeration to say that Spock is hardly in the film.
I had that realization the last time I watched it. I had always known it was essentially Kirk's story, but it finally occurred to me how little Spock really did in it. That may be because as the story goes, the death scene was actually written into the script much earlier in the story, then kept getting pushed back to increase Spock/Nimoy's time on screen and give him a bit more to do.
In TVH, one thing that may go unnoticed by young'uns watching (even if they see it) is that on the bus, after Spock neck pinches the punk, you can see one of the passengers is reading an issue of "Omni" magazine. "Omni" was a magazine in the 1980s that dedicated itself to high-quality science and science fiction writing. Kind of like the "Atlantic Monthly" of science. You can imagine how thrilled a reader of that magazine would've been to know two men from the future were on that bus with him.
Not so much an observation, but one thing did occur to me for the first time watching TVH this weekend. Kirk says they've been on Vulcan for three months, yet they are all leaving Vulcan wearing the exact clothes they wore at the end of TSFS. They must've really liked what they had on.