At time stamp 2 minutes 21 seconds Captain Batel is wearing a USS Cayuga badge in her left arm and at 4 minutes 3 seconds it has changed to a USS Enterprise badge.
As Chapel and Spock meet outside the wreckage of the Cayuga, in the wide shot, Spock's phaser is in his holster, but in the close-up, it is gone.
The red alert Klaxon is still going off in the ship long after it was started. The Klaxon is meant to alert the crew that a state of emergency has started. Once all crew are awake and at their posts the Klaxon would be turned off (because its loud and annoying) though the red alert lights would stay on.
Ortegas says that the hunter ship is "in the fourth Lagrange point" but all exterior shots and screens on the ship show that it is nowhere close.
Near the end, Pike gives the coordinates of the colonists as "2 km south by southwest". This is a mistake, no doubt influenced by the North by Northwest festival. The correct wind rose direction is southwest by south (the compass point halfway between southwest and south-southwest, specifically 213.75°, abbreviated as SWbS).
Chapel bangs her hand against the window and calls out for Spock. Since he is in space, there would be no way for him to hear her. She should have waved her arms in front of the window to attract his attention via movement.