Uhura says that the shuttle can't contact the Enterprise because interference from the spatial vortex limits their comm range to one light-year. However the Enterprise would have to be much closer to the moon than a light-year, as class F shuttlecraft carried on Constitution-class starships in the mid 23rd century were only capable of sublight speeds. A class F shuttle was only equipped with thrusters and impulse engines, at full impulse they would be traveling at .25c (one quarter the speed of light), and at maximum impulse 0.5c, a one light-year trip would take six months. It wasn't until the mid 24th century that Starfleet's warp propulsion technology had advanced enough to make a warp core small enough to fit in a shuttlecraft.
The knit cap that Spock wears to conceal his human features from his mother has an improperly orientated Starfleet delta emblem on it. The emblem is mirror-flipped from how it should appear. (In fairness, this might have been intentional, suggesting that Spock hastily put the emblem on wrong and had no time to change it.)
Spock mind-melds with his mother Amanda. However, in Dagger of the Mind (1966) which takes place chronologically 7 years later, it is a large plot point that Spock has never Mind-Melded with a human before. This is a fact that was clearly overlooked or ignored by the writers.