Continuity mistake: The Bird of Prey is the one captured by Kirk's crew in ST III That ship's bridge showed Klingon Cmdr Kruge in his elevated command chair with his helmsmen arrayed circularly below him, and nothing else. ST IV has this same ship, however the bridge now resembles The Enterprise layout with Kirk's command chair behind Sulu and Chekov at their rectangular helm, with Spock, Scotty and Uhura at their usual positions.
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)