Likes:
Cinematography. Those scenes on Earth were utterly convincing.
The way Khan was used. I hoped and prayed Khan wasn't the villain in this film. I still wish they had come up with something more original, and to an extent, they did. Marcus was the enemy and Khan was his tool. That business with shooting drones (I'm sorry, torpedoes) at one citizen (I'm sorry, fugitive superman) without due process echoed troubling contemporary news. In your face, Mr. President! In the Star Trek universe, if you do this you are one of the bad guys, and Kirk and Spock will whoop your butt!
The actors. I didn't like Simon Pegg as Scotty in the last film, but he was great in this one. He was, like Scotty in the original series, the ordinary man with whom the audience can relate as Kirk and Spock do really insane things. Karl Urban again nailed McCoy ... the man has Kelly's Katra in a jar somewhere, I'm sure. Pine was great as Kirk ... he's adopted a bit of Shatner's swagger while bringing his own touches to the character. And Quinto ... was his voice a bit deeper this time? I'm perfectly happy with this guy being Spock for as long as Nimoy. Just ... give him some prosthetic earlobes.
The music. In 2009, Michael Giacchino delivered a soundtrack that struck my ears as a bit dull and repetitive. This time, there's a lot more variety to the music, with frequent callbacks to Alexander Courage. I'm looking forward to adding the new soundtrack to my collection.
The wardrobe. The costuming was gorgeous! I love the callbacks to the original series and Star Trek: The Motion Picture. I also like those jackets with the clear windows showing the color of the tunic underneath. The costume designer is a genius with a great eye for detail and design.
Dislikes:
The Enterprise tumbling in free fall while its crew is falling inside the ship. Any film-maker who has been paying attention to the biggest films of the last twenty-five years must have known how Apollo 13 was made. How the sets and actors were hauled up in NASA cargo planes that then performed a series of free-fall manuevers while the actors performed in weightlessness. Why then did everyone in the free-falling Enterprise ... fall? They should have floated. They were falling in the same gravitational field as the ship. This was the biggest gaff in the movie.
Empty space around the Earth and Moon. Isn't Earth the center of the United Federation of Planets? Why did no other starship assist when the Vengeance started pounding on the Enterprise while only 237,000 kilometers from the capital planet of one of the most powerful empires in the galaxy?
Engineering. Nope ... still a brewery. I liked the warp core scenes shot at the National Ignition Facility a bit better, but the hardware still looked too contemporary. This has been complained about since 2009, but no one is budging from their position.
Too much junk on the Bridge. Another complaint from 2009 is the way a perfectly beautiful interpretation of the Bridge has been ruined with too much clutter from the 21st century and too many lights.