After Kirk is given the adrenalin injection and he is shown writhing, his trousers change colors from blue, to black then back to blue.
Upon beaming down to the planet surface in the first shot, the buildings are round, with alternating light blue-green and white panels. After a few shots and Kirk, Spock, and McCoy discussing Kirk's previous conversation with Dr. Robert Johnson, the camera is trained on the "ribbed" building again, except now the alternating panels are light plum or purple and white.
Lt. Galway's hair when she's old is different.
As Act One begins, Kirk's voiceover reads the Captain's Log while the Enterprise leaves orbit and heads into space. Minutes later, in the Briefing Room, Kirk tells the assembled officers that he'll keep the Enterprise in orbit until an explanation for the aging sickness can be found.
Early on, Commodore Stocker consistently addresses Captain Kirk as "Sir", even though it has been established in previous episodes (and in naval protocol) that commodores outrank captains. However, "sir" is also used as a means of polite (or impolite, depending on intonation) address in conversation. Spock and Khan, for example, exchange "sirs" in the banquet scene in Space Seed (1967), and Kirk, McCoy, Hengist, and Jaris trade "sirs" among themselves in Wolf in the Fold (1967).
Even if there were a disease that leads to such a rapid aging, there are a few biological laws. It is impossible for a person's hair to suddenly whiten (or later return to its original color) because the hair itself is no longer part of the systemic circulation. However, this is science fiction and could refer to strange phenomena not yet understood. Moreover, it is possible for white or grey hair to regain its pigmentation. It is a rare phenomenon called 'anamelanism'.
Kirks hairline is inconsistent. When he first starts aging, the hairline is receding but, as he gets much older, his hairline advances again.
In the final shot, Chekov is suddenly wearing a wig instead of his natural hair. (The shot was obviously borrowed from an earlier episode.)
The cure for the rapid aging disease not only restored hair color and smoothed out everyone's skin, it also was able to restore hairdos! After Kirk and McCoy are both cured, their hair returns to the hairdos they had at the start of the show.
At the start of the competency hearing, Spock states that Captain Kirk is entitled to question the witnesses after the board has finished examining them. He then questions four witnesses as well as the ship's computer without ever giving Kirk an opportunity to do so. He even dismisses one of the witnesses from the hearing immediately after her direct examination.
After the Enterprise crew's experience with a virus in Miri (1966), their lack of concern about beaming up the infected landing party seems odd.
Dr. McCoy states that the landing party is aging at the rate of 30 years per day. Spock then states that the landing party only had, at most, 7 days to live. At the rate that Dr. McCoy mentioned, the landing party would be in excess of 210 years old physically when they died. What if Spock just calculated for how long he had and applied it to the rest of the landing party?
Spock is guilty of leading one of his witnesses (putting words in their mouth, one definition) which is not permitted in court. When he questions Lt. Uhura about Captain Kirk forgetting about the Romulans had broken code 2, he replies, "Yes, but he changed it to code 3." Spock then says, not in question form, but as a statement "After you reminded him that The Romulans had broken code 2." Uhura submits and says, "Yes." It was up to Uhura to testify that she reminded him, not for Spock to tell her that she reminded him.