Continuity mistake: After Garth attempts to convince Mr Scott that he is Captain Kirk and to beam him up, while he is banging his fists on the floor in anger he transforms back into himself and the fly like ring on his finger breaks and slides out of shot. It's quite hard to see but it does. The next time we see Garth's hand his ring is intact yet he has not transformed himself again to repair it.
Other mistake: At the end, Kirk teases Spock twice specifically about letting himself be hit on the head in order to determine which Kirk was genuine. But the phony Kirk never hit Spock on the head. He merely pushed Spock over and attacked the real Kirk. Clearly, the action sequence wound up differing from the scripted dialogue, and no one noticed. (Also clearly, Kirk is not speaking metaphorically here.) (00:45:30 - 00:49:20)
Other mistake: The only way to beam up to the ship is to repeat the chess counter move after ASKING to be beamed aboard. After the two Captain Kirks fight ends with Spock stunning Garth with the phaser, Spock contacts the Enterprise. After saying "Spock to Enterprise, and hearing Scotty say back to him "Scott here", Scotty says" Queen to Queen's level 3" without Spock requesting to be beamed aboard or even waiting to see what Spock wanted in the first place. (00:46:56)