Continuity mistake: During the scene where Kirk is talking in the briefing room, the director was obviously not pleased with how James Doohan said his lines. Due to budget constraints, instead of re shooting the scene, they simply replaced Mr Scotts dialogue with another one. You can see it since at one point his lips are completely out of sync, even on the remastered edition.
Continuity mistake: The landing party had been apprehended and placed in a locked room under the supervision of a single guard. Spock manages to free them and Kirk knocks out this guard, leaving him in the locked room. Later on in the episode, they return to this very room and somehow this guard is gone. He could not have woken up and walked out because it is a locked room and Kirk took his key.
Factual error: Spock presents himself to Dr. McCoy in sickbay to be relieved of duty for "killing" Captain Kirk. However, as a medical officer, Bones has no place in the ship's chain of command. Doctors have an officer's rank in recognition of their education and value to the service, but no authority over military matters. (For example should Kirk, Spock and Scotty be indisposed, command of the ship would fall to Sulu who is in the chain of command even though McCoy has a higher rank) If Spock believes he should be relieved for breaking regulations and committing murder, and thus confined until court-martial, Scotty would be the one to do it, as he is the next senior officer. And this is not a character mistake, Spock quotes regulations regularly, there is no way he'd ignore chain of command. Contrary to what viewers seem to think, militaries have regulations and officers don't get to ignore them and do what they feel like! The only reason Spock talks to McCoy is so Kirk can walk out of the back of sickbay and create a nice emotional moment.
Continuity mistake: Kirk and Spock pretend to be filming Daras so they can enter the building where John Gill will give his speech. Look at the lamp Spock is holding. In the beginning it works, than a couple of scenes later it doesn't, and when they are outside the chamber where John Gill is, his lamp is on again.
Continuity mistake: After his initial mind meld with the horta, Spock tells Kirk, "That's all I got, Captain: waves and waves of searing pain." A minute later, he says that it's "a highly intelligent, extremely sophisticated animal" that calls itself a horta. Apparently, waves of searing pain were not all that he got after all. Unlike Spock to be so imprecise.