Character mistake: When Spock beams down after calculating there was just enough energy left for him to do so, Sulu and Kirk watch him materialize, and Sulu says, "Someone beaming down from the bridge." From the bridge? Shouldn't he have said "from the ship"? No one ever beamed down directly from the bridge, but even if they could, how would Sulu know that's where they'd come from?
Character mistake: In the scene when the shuttle is finally repaired and lifting off the planet, Spock asks "Fuel status?" and Scotty incorrectly answers "15 pounds PSI, sir. Approximately enough for one orbit." The correct answer would be simply "15 PSI" without the redundancy, as the P in PSI already signifies pounds. As an engineer Scotty would know this.
Character mistake: Spock expresses amazement that the natives of this planet speak Earth English, however, he and Kirk were both on the bridge when Uhura tapped into the planet's news broadcast and already know that they speak Earth English, and there's no reason to tell McCoy, because he heard it too. Commenting on it would be illogical.
Character mistake: Spock tells Kirk that without being able to test their vaccine with the ship's computers, "it could be a beaker of death." The vessel that the vaccine is contained in is a flask, not a beaker. The ship's science officer should know the difference.