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.
Plot hole: Spock has absolutely no way to know, yet, that the horta only secretes her corrosive substance when tunneling: he hasn't had time to examine her or to do more than determine that she does indeed secrete a substance that cuts the tunnels. So he should at the very least scan the piece of her that falls off before he picks it up with his bare hands. Major lapse of logic, which given his character goes beyond a character mistake.
Deliberate mistake: Here, as in nearly every other ST episode featuring caves, caverns or mines, all the floors are perfectly flat and the lighting adequate-to-see-by or even brilliant. These configurations are easily found on soundstage sets - but not in nature. (00:16:30)
Revealing mistake: The Horta's round tunnels have perfectly even striations and are obviously factory-manufactured tubes. Acid secreted by an elliptical creature burning through solid rock would not create a perfect circle. The Horta is visibly not chewing, sculpting or smoothing the sides. No acid burning method would leave patterned stripes on the walls, either. (00:18:15)
Character mistake: Kirk informs Spock via communicator the Horta is ten feet from him, and Spock insists Kirk kill it. First, both know they cannot kill it with their phasers, and second, Spock's demand for Kirk to kill the Horta runs counter to the Vulcan philosophy of respect for all life. Spock would never want to harm, let alone kill, another life form.
Continuity mistake: As Kirk is crawling through the tunnel leading to the Horta's nest, a tight view from his left side shows light shining on his face and darkness behind him. When the shot changes, though, to the view from the nest, there is light behind him and darkness on his face.