Factual error: When Sam speaks with his daughter on the long range phone there is virtually no delay between some questions and answers. It takes over two seconds for radio waves just to travel from the Moon to the Earth and back in a straight line. The interval between question and answer physically can't be shorter than that and realistically should be around 3 seconds or even more, accounting for additional signal relays and time needed for a person to form a statement. Same for the communication between Gerty and the Lunar employees when Sam 2 got out of his infirmary bed.
Continuity mistake: When Sam 2 places Sam 1's helmet on his head before placing him back into the damaged rover, the faceplate is still blood-spattered and dirt-smeared, obscuring Sam 1's face. However, when Sam 1 is settled into the seat in the rover, the faceplate is very clean and his face is visible.
Factual error: The Earth and the Moon are not in the same phase. The Earth is gibbous, while the Moon is crescent. In the real world, they should be in almost exactly the same phase from every angle of view. (00:01:50)
Continuity mistake: In the scene where Sam crashes in the vehicle, a drip of blood appears just below his right eye (the viewer's left). However, in the next shot, the drip moves to his left.
Deliberate mistake: When Sam is watching the security footage of his predecessors being disintegrated in the chamber, the disintegration happens the instant the coffin door is shut. However when he goes in to test it himself there is a ~60 second wait as a congratulatory video plays before the disintegration occurs. (01:00:00)
Continuity mistake: When Sam B knocks the models over and they start fighting, Sam A is knocked to the floor. As he's trying to get up you can see a cut and quite a bit of blood on the top of his right hand because his bandage is gone. A second later, as he's standing up, the bandage is back. (00:47:00)
Continuity mistake: During the opening credits, as Sam Rockwell enters the rover there is a piece of equipment placed next to the hatch. It is not taken inside by Sam, as the next shot shows him climbing down empty handed. The apparatus however mysteriously disappears in the next shot of Sam closing the hatch.