When it comes to that "Star Trek Feel" of the classic series and movies, the most obvious absence is the Kirk and Spock relationship. In the original series, there was no doubt that Kirk and Spock were good friends, but it was never done in such a way where every situation the characters were faced with had to have some form of impact on their friendship. The two spent a lot of time figuring problems out without being in each other's faces about whether or not it was the right or wrong thing to do. That's not to say every episode should be like this. Amok Time has Kirk really pulling for Spock in order to get him home to aid in his ailment, and "The Menagerie" has Kirk dealing with Spock who disobeys orders and steals the Enterprise. There are good episodes that showcase these types of conflicts, but they weren't the routine. With these new movies, EVERYTHING that happens has to affect their relationship is some epic way, even on things that in the original series they would just wave their hand at. Violate the prime directive? Inexcusable! Going after Khan? You should know better!
This is what happens when you have writers who take this material for granted rather than what it is. They literally write lines of dialogue telling you that their relationship is special instead of, you know, just making their relationship special. And some of the best relationship moments between these two, or heck, all the characters, are the things that subtle and rare. With the original characters, they were friends but they acted professionally. With these new characters, it's ALWAYS about their friendship.