Other mistake: During the inquiry at the start of the movie, the Klingon Ambassador is going over the footage of the destruction of the Enterprise from all the exterior views. This is very nice, but how would they have access to all of these different viewpoints of the destruction? Bearing in mind no-one was there to film it. It couldn't have been the Klingon Bird of Prey that filmed it, because in most of the destruction scenes, the Bird of Prey would have been no-where near or in a position to film it, and was of course captured and still being commanded by Kirk during the course of the inquiry. (00:04:10)GalahadFairlight
Other mistake: When we first see the Klingon bird of prey on Vulcan (just before we see closeups of the "Bounty" written on the side) it is clear that the ship is far too small to contain a pair of adult humpback whales plus 18000 cubic feet of water in its hold. Later, when the ship is hovering above the sea after foiling the whaling ship's attack it seems considerably larger.