Factual error: It is long-established in Star Trek canon that onboard diagnostics can detect any animate intruders on Federation vessels. Any living thing that exists upon a Federation vessel can be identified, and its location specifically noted on Federation property. How is it, then, that there are rats aboard the Regula I space station (as observed by Doctor McCoy) that haven't been eradicated?
Continuity mistake: When the Enterprise is making a run for the Mutara Nebula, they show a scene in a photon torpedo bay. On the wall the sign says "TORP BAY 4." The problem is that the Enterprise only has 2 photon torpedo bays. Bay 1 (Starboard side) and Bay 2 (Port side). This version of the Enterprise has no aft photon torpedo bays.
Continuity mistake: When Admiral Kirk and party arrive in the transport pod, they dock on the port side docking port in the secondary hull. But the airlock they emerge from is in the docking port for the port side torpedo bay, as signified by the sign in the background saying "TORP BAY 2." You can see the docking port they are supposed to have docked at to emerge from on the side of the torpedo launchers, just above of the secondary hull. The reason for this is because they re-used the stock footage from ST:TMP when Kirk and Scotty docked at the same docking port.