When the Enterprise approaches Talos IV, there is a close-up of Lt. Hansen sitting at the helm. In the next wide shot of the bridge, Lt. Hadley occupies his place.
When Number One is in the transporter room, she is wearing a bright shade of red lipstick. Seconds later, when she materializes in Captain Pike's cell on Talos IV, her overall makeup is lighter, with a much lighter lipstick, matching Yeoman Colt's shade.
When fighting the Kaylar in the illusion of Rigel 7, Pike holds the Kaylar off with a staff as he backs up the stairs. The staff appears to have a metal blade. When the blade is pushed against the Kaylar's body, the blade flexes like rubber.
When the Enterprise landing party is using the laser cannon to blast through the Talosian's entrance, multiple studio lamps with steady illumination are reflected in their goggles. But with camera(s) facing the target, the fluctuating light of the laser cannon is seen.
When the illusion of Commodore Mendez shimmers away, the angle of the chair that is left behind does not match the position in which the superimposed image of Mendez was sitting. Mendez was leaning off to his left, yet the chair back is angled to what would have been his right. The positions of his arms also do not match the locations of the arm rests.
If the Talosians want to create a new race of slaves, they ignore the fact that the race that Pike and Vina would produce would be extremely inbred.
When the Talosian tries to get the weapons by sneaking through the door of the cell, Captain Pike pulls him in. When the illusion is removed and reveals the hole in the glass wall, the group leaves through the hole. They could have just gone out through the door of the cell since it was open.
When the Enterprise reaches Talos IV, Spock informs Kirk and Mendes that "The Talosians control the Enterprise now, just as they did 13 years ago." At no point in the archival footage showing the events of 13 years prior were the Talosians in direct control of the Enterprise. At best they threatened to control the minds of its crew to make them operate the controls incorrectly and destroy the ship.
When the Enterprise crew fire the phaser cannon at the rock door, Number One gives orders through the communicator to the Enterprise, which is providing the power for the cannon. When Spock reports the circuits are overheating, she forgets the communicator and shouts "Disengage!" up at the sky, as if the Enterprise could hear her like that.
Number One's briefing room comment "So we just THOUGHT we saw survivors there" should not have been in the first person plural (we), since Number One herself was on the Enterprise during the referenced incident.