Plot hole: In the conference room, Spock informs Kirk that his theory of a Romulan cloaking device kept the sensors from detecting the ships. However, in S1:E14, "Balance of Terror", they figured out the cloaking device, and how it worked. Unless he was feeding into the deception, Spock shouldn't have had any theory when it was a fact.
Continuity mistake: The scene outside shows a guard standing close to a sign on the wall that says, "Jo Jo Krako", but the next overhead shot shows the guard further away, and a railing between the sign and the guard that would have prevented anyone from getting close to it.
Other mistake: On the bridge, as the camera looks at the view screen, there is a flashing blue light reflecting on the console in front of Sulu, but there is no light other than scanner under the viewer flashing and it's too far away to project light that far, and it's amber colored.