Plot hole: The ambassador and his aide beam down to the planet. This would not be possible because earlier on Scotty refused to lower the screens/shields until the Captain told him to do so. It had already been established in an earlier episode titled "Arena" that transporters don't work with screens up.
Plot hole: The law officer who arrests Kirk recalls hearing Kirk call the "spirit" Bones. But he wasn't there when Kirk spoke to McCoy. He rushed into the scene several minutes later. If he's lying, how would he know that Kirk used the name Bones? He wasn't there to hear it. Even if he had been, Kirk did not refer to McCoy as Bones in that conversation.
Plot hole: It is stated during the episode that Khan was, at one time, the ruler of over 1/4th the Earth's population, during a very key moment in the planet's history (The "Eugenics" Wars). Such a personage undoubtedly would be very well known to 23rd century Earth people, at a level of infamy approximating Julius Caeser or Adolf Hitler. Yet it is only 2/3rds the way through the episode, thanks to a computer search by Spock, that the crew divines his identity. Lt. McGivers at the very least should have almost instantly recognized him.
Plot hole: When Captain Kirk is the last person unaffected by the spores, he's on the bridge lamenting the fact that he is trapped in orbit above the planet since he can't pilot the ship alone. He also can't call for help because Lt. Uhura sabotaged long range communications. but Captain Kirk has somehow forgotten about the shuttle craft. He could have easily escaped on one of those.
Continuity mistake: While Kirk questions Spock about his command decision on the Galileo, shots of Uhura and Mears show the ladies close together at Uhura's station with Mears' hands hidden but close up shots of McCoy show Mears and Uhura in the background further apart from each other and Mears' hands in front of her body.
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.