Audio problem: When the Enterprise fire the torpedo, the sound used is the old series sound used when the ship's phasers were fired, not the correct noise for a torpedo firing.
Other mistake: Spock lifts his crew mates with the rocket boots. He passes deck numbers 35 through 78 from bottom to top. First off, deck numbers go from top to bottom. The bridge is on deck 1. Second, the Enterprise of that class only had 23 decks.
Uhura: Bad news gentlemen, shore leave has been cancelled.
Chekov: Rescued at last!
Uhura: Return to pre arranged coordinates for pick up.
Sulu: Coordinates?
Chekov: Don't tell them you're lost, you'll never live it down!
Uhura: Is there a problem, gentlemen?
Sulu: Uh, yes, uh... We've been, uh, we've been caught in a blizzard!
Chekov: [Blowing into the communicator.] And we can't see a thing! Request you direct us to the coordinates!
Uhura: My visual says sunny skies and 70°!
Chekov: Sulu look! The sun has come out! It's a miracle!
Uhura: Don't worry fellas! Your secret's safe with me. I'll send a shuttlecraft to pick you up.
Sulu: Uhura, I owe you one. Sulu out.
Trivia: If I am correct the Yoman who gives Kirk the status report he signs on the bridge is in reality William Shatner's daughter, Melanie Shatner, who had also appeared in TOS Episode Miri as one of the children.