They tried to copy the Spock/Kirk dynamic throughout to make that reverse copy of the death scene from TWOK work, except they didn't have 90 hours of TV showing established friendship to back it up with any substance. It just fell flat as an empty parody of a much better movie. "I hated you two hours ago but now we're BFFs."
They tried to copy Khan as a villian, except without the backing of a personal vendetta growing over 15 years and a pre-existing knowledge to give it any substance. It also fell flat because they didn't know who the hell he was. Cue crickets sound when he reveals to them his real name. Ok, your name is Khan. That's very interesting Mr British White Male.
They introduced Carol Marcus, except with no relation at all to what her character was to Kirk or TWOK, her only purpose was to be objectified as eye candy, and possibly as a red herring to throw off fans as to which TOS story they were copying (same goes for Khan).
No, the movie isn't a direct copy of TWOK, but official remakes often aren't either. I've seen remakes that borrowed less from the source material. They take the key elements, and create something new from it. I like the way hef put it. It's an inversion/perversion of TWOK.