Just before Kirk and Spock stun Mitchell (just prior to beaming down to the planet), the shadow of the boom mic is clearly visible well within the frame, moving across the wall behind Spock as Mitchell gets out of bed.
When the Enterprise officers are in the boardroom discussing the increase in Gary's ESP abilities, Sulu uses the word "geometric" instead of "exponential". While both refer to growth rates of a mathematical function proportional to the function's current value, "geometric" refers to a function with discrete domain values; "exponential" refers to a continuous domain value. (Sulu's description of starting with a penny and doubling it daily is correctly geometric, but Mitchell's ESP was continually growing, so exponential would be more appropriate.)
During the chess game in the beginning, Spock mentioned that one of his ancestors married a human once. Although technically correct, as later on his mother is revealed to be human, it would be illogical to mention it in this way. This indicates that originally Spock wasn't written as half human, but only slightly human, perhaps 1/8 or 1/16. Just enough to irritate, as Kirk taunts Spock with it.
When looking at Dr. Elizabeth Dehner (Sally Kellerman) profile, she is listed at 5'2", while Lt. Cmdr. Gary Mitchell (Gary Lockwood) is 5'9". When they stand next to each other though, they are close in height. Sally Kellerman is actually 5'10" and Gary Lockwood 6'0 1/2".
As Kirk, Spock, and Mitchell enter the bridge from the turbolift, a black crewman wearing a blue shirt crosses behind Yeoman Smith and has entered the turbolift by the time Kirk reaches the steps down to his chair. In the next shot, however, the same crewman is still walking toward the turbolift when Kirk reaches the same spot. In addition, Yeoman Smith is carrying some sort of cards close to her body when Kirk, Spock, and Mitchell exit the turbolift; in the next shot, the cards are gone and her arms are at her sides.
When Spock first reports on the information from the Valiant's buoy, he states that the tapes are burned out, and that he is going to begin reading the memory banks (electronic). Then a minute later, he states that he's having trouble reading the tapes.
There is a struggle with Mitchell before is made unconscious and it is obvious that he doesn't have an undershirt on. For the rest of the show, he - and his stunt double - are seen to have an undershirt.
As the Enterprise crew prepares to enter the Galactic Barrier, Kirk says to Mitchell to get ready to leave the galaxy. Even at maximum warp speed, the Enterprise would not have made it to the edge of our galaxy in such a short time. Therefore, Kirk's statement would be false. Further, in "By Any Other Name", Kirk tells Rojan that it would take thousands of years to reach Andromeda, the nearest galaxy to ours, at maximum warp. Then, Rojan tells Kirk that their modifications would only take 300 years to reach it. Any way you look at it, leaving the Milky Way galaxy is not possible with the technology afforded.
After the initial attack Kirk states that without Warp Drive they are now years not days away from human colonies. The difference would be 10's to 100's of thousand's of years, going from many times the speed of light to sub-light speed.
The events of this adventure obviously occur significantly earlier than the bulk of the rest of the series. As such, the educational title of physicist that is given to Sulu is in line with subsequent appearances where he is later promoted to the positional title of, and referred to as a 'helmsman'. The same applies to Spock's different rank and uniform color as well.
On Delta Vega Captain Kirk says he wants someone with Mitchell and someone watching from "the dispensary". But Delta Vega is a fully automated station where the ore ships only stop "every 20 years", so it makes very little sense the station would have a dispensary.
When Kirk and Spock are about to take the elevator to the bridge, an overhead view of the entrance to the lift is seen where the floor of the lift and the deck of the corridor are obviously continuous.
At one point while he is in the sick bay, Gary Mitchell is shown rapidly flipping through computer files, supposedly to indicate that his ability to absorb information is increasing exponentially. Close examination of the screen, however, shows that the same picture is shown on the screen over and over.