Other mistake: Taylor believes he is on a planet 300 light-years from earth for most of the movie. However, after spending months there, he would have seen the moon at night a number of times, letting him know where he really was before seeing the Statue of Liberty.
Suggested correction: Not necessarily; any astronaut knows that many, many planets have moons, so Taylor could have easily thought that it was simply an odd coincidence that the "other" planet's moon closely resembled Earth's moon.
Other mistake: As the Apes first appear on horseback, as the second one from the camera's left turns around, you can see that the rifle doesn't have an ammo clip in it.
Other mistake: After the rocket hits the water, the view of the landscape bobs up and down. But the movement at the top and bottom of the view are slightly jerky rather than the smooth stop and go that a real bobbing craft would have.
Other mistake: After running from the falling boulder in the forbidden zone, Dodge walks ahead of Taylor and Landon. In one shot, Dodge is shown in the distance, but you can see two tracks in the dirt between him and the others. (00:19:50)