Continuity mistake: Gillian's Chevrolet pick-up truck is obviously two different trucks, depending on its location. When near the aquarium, the grille is a series of large, rectangular openings. But when at the park, where the Bird of Prey landed, the grille is a much tighter series of more numerous, smaller rectangles, made up of much thinner bars. This difference is consistent through the entire film.
Factual error: When Chekov and Uhura first see USS Enterprise and then later beam aboard, the carrier shown is not USS Enterprise (a nuclear aircraft carrier), but rather USS Ranger (a conventional or oil-fired carrier). The most obvious clue is the "island", the huge structure that sits on the deck of the carrier. USS Enterprise has the most distinctive and unique island of all the USN carriers, being totally cube-like in appearance. The carrier in Star Trek IV has a more conventional style island. [It is true that the USS Ranger was used in lieu of the USS Enterprise for the movie. This is because the Enterprise was on deployment at the time. Though by the time the movie was made the Enterprise had had its distinctive pagoda-like island rebuilt by Puget Sound Naval Shipyard in 1980, the aircraft carrier served until 2012 with a still-distinctive cube shaped island.]
Continuity mistake: When the 'Bounty' materializes right above the whaling ship, it casts it into shadow. But the close-ups of the captain and navigator as they frantically turn the ship around reveal no trace of a deeper shadow on their faces and the surrounding waters; the lighting remains the same as in the scenes when they try to catch the whales.
Continuity mistake: In the scene where Kirk and Spock are walking down the street after Spock has been in the tank with the whales it is clear that it is a windy day. Both of them have their hair blowing in the wind and yet as they are offered a ride in the truck and the shot leaves them for a moment their hair is suddenly fixed again.
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.
Plot hole: When Spock analyzes the call of the probe and compares it in a phylum search, he's presumably interfacing with the Federation memory bank as mentioned in Sulu's pre-launch status. However, the alien probe was disrupting all power and communications in the vicinity of Earth. It's extremely unlikely the ship could've interfaced with the Federation memory bank at the time of that search.
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.
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.
Visible crew/equipment: At approximately 01:32:50 on the Blu-Ray, you can see a diver releasing the mechanical whales. He is on the far right side of the screen, you can make out his head, body and left hand. He is visible for a good 8-9 seconds. If you have a different media source, it's during the Bird of Prey decent. You see a closeup of Kirk, followed by a closeup of Uhura and Gillian, then a close up looking down on the harpoon. It's in the following scene.