Other mistake: When Sulu tells Kirk he is coming to help, he states the Excelsior is currently in the Alpha Quadrant. The problem is the the Alpha Quadrant in the Star Trek universe is one quarter of the total Milky Way Galaxy. While technically correct, Sulu giving his position as in the Alpha Quadrant is useless to Kirk in terms of knowing when to expect help. It doesn't matter where Khitomer is in the galaxy. Sulu saying he is in the Alpha Quadrant gives Kirk absolutely no idea of when he will arrive. He could be anywhere from 5 minutes away to 5 weeks away or even further. Remember, it was originally projected to take Voyager 70 years to go from the Delta Quadrant back to the Federation in the Alpha Quadrant.
Other mistake: It was shown that whatever form Martia was in, whether it be the attractive semi-human form, the Sasquatch form, or the little girl form, her voice always remained the same. Why didn't her voice remain when she morphed into Kirk?
Other mistake: Every time we see a high ranking Starfleet Officer in the Starfleet Command scenes, they are wearing dozens of different badges and medals. Why don't Kirk and crew have any? In just the movies, Kirk and Co. have saved Earth twice and foiled an internal assassination attempt against the Federation President, as well as all the heroic and successful missions carried out in the original series. In the classic episode "Court-Martial" it lists Kirk as receiving the Starfleet Medal of Honor, the Silver Palm with Cluster, Starfleet Citation of Conspicuous Gallantry and Kragite Order of Heroism. And it is appropriate to Starfleet regulations to wear them. Kirk was wearing two decorations in the TOS episode "Court Martial" when he was framed for a crewmans' death.
Other mistake: Throughout the film, Valeris is referred to as a Starfleet Lieutenant. However, in every scene, she wears the rank of Commander on her uniform. Best comparison can be made in Star Trek II: The Wrath of Khan where the same uniforms were used. Saavik (Kirstie Alley) wore a Lieutenant rank (2 piece), where Chekov and Scotty (himself then still a Commander, until promoted by Admiral Morrow in Star Trek III) wore the single-piece rank insignia used by Valeris in The Undiscovered Country.