TOS was very much a show based in the 60s, in terms of tone and format, with the differences coming from the casting. It was pitched as an action/adventure show, with some social commentary, as science fiction often does.
21st century filmmaking tends to be much faster paced, with an emphasis on action. And, that is certainly present within the Abrams films, but there is definitely a strong line of social commentary built within it.
As for TOS elements there is a
video on it but I'll break it down real quick:
-Sarek's speech to young Spock regarding the nature of Vulcan emotion from the Animated Series, as well as the teasing by Vulcan youngsters.
-Spock and Uhura flirting in one episode.
-McCoy's comment of "Little suffering is good for the soul."
-Red shirt on the landing party.
-Kirk goading Spock in to an emotional outburst.
I could go on, and the video shows different clips side by side to illustrate further. But, my larger point is, that Abrams Trek is true to the spirit of TOS, not later Star Trek. It focuses on the action and adventure of space, and weaves social commentary around that.
People's experience obviously will vary, but I think it works very well.