Revealing mistake: Most of the posed, fake displays in the ape's Museum of Natural History contain real people who are slightly moving if one pays careful attention. Especially noticeable are those posing with objects held in - or above - their hands.
Continuity mistake: When the spaceship first springs a leak and begins to sink, the interior shots show it tilting to starboard; Taylor falls against the starboard bulkhead and the other two survivors need to brace themselves from slipping. All of the exterior shots of the ship show it level from port to starboard.
Continuity mistake: The first exterior shots of the spacecraft in the desert lake show the spaceship's hatch door already gone... before the astronauts blow it off with an explosive charge minutes later in the movie.
Continuity mistake: At the end of the movie when Taylor and the woman find the ancient ruins of the Statue of Liberty, their distance from the statue seems to change between shots. In the first shot (when it only shows them and the torch) they seem to be almost next to it. In the next shot (when it only shows them and the crown) they are farther away. And in the last shot (when it shows Taylor kneeling from behind and the full torso of the statue) he is even farther away from it.
Continuity mistake: When Nova is removed from Taylor, one ape is also soaked with the hose. Then he is totally dry while taking Nova out of the cage.
Revealing mistake: Toward the end of the film, in a long-shot, an ape hits another in the head with the butt of his rifle, knocking him over. You don't even have to look closely to see that the rifle's butt never even comes close to making contact with him.
Revealing mistake: Towards the end of the movie, at the cave, you can see that the rifle Charlton Heston has been firing has no magazine inserted.
Continuity mistake: Shortly after the astronauts crash land in the desert lake, they discover that the female crew mate is dead, her body is mummified due to an air leak in her cubicle. The mistake is that when her body is first seen, the arms and hands are folded across her chest. Seconds later, when the water starts rushing in, the point of view from inside her cubicle shows her arms at her sides.
Visible crew/equipment: When the astronauts are running down a dusty hill in the desert, you can spot the shadow of the camera on the back of Landon and Dodge. (00:15:50)
Revealing mistake: Throughout the movie, if you look closely at all of the apes' mouths, you can see the actors' teeth behind the ape teeth.
Continuity mistake: When Taylor, Landon and Dodge discover the humans, when Dodge says "Blessed are the vegetarians", both Taylor and Dodge are standing even with each other and Landon is standing about an inch or two behind Taylor and Dodge. In the next shot, Landon and Dodge are standing even with each other and Taylor is standing almost a foot away from Landon and Dodge. (00:29:00)
Continuity mistake: Taylor, Landon and Dodge go for a dip in the river and they get wet. The humans then take their stuff, so they get out of the river and they are all mostly dry (still a bit damp, but less dry than they were before). (00:25:40 - 00:27:25)
Continuity mistake: When the three are doing water check, the wide angles show Landon sitting in the sun, but the closer angles show Landon sitting in the shade. (00:17:15)
Continuity mistake: After the ship has sunk, Taylor, Landon and Dodge make camp on some rocks. In the shot where Taylor says "Dodge, run your soil test. Got your sensors? Geiger counter?" in the background is the life raft used when they escaped the ship. The raft is resting on some rocks that are bright colored. In the next shot, in a shot of Landon the rocks the lift raft is resting on are all dark colored. The raft has also moved closer to the water. (00:12:35)
Continuity mistake: Towards the end of the movie when Taylor, Cornelius, Zira, and Nova are about to enter the cave, Zaius arrives with his crew. Taylor comes running back taking a few shots at the soldiers then he points his gun at Zaius. In a frontal shot you will notice the gun he is pointing has a strap on it tucked under his left hand as he is aiming. The scene goes to a rear shot of Taylor in the foreground and Zaius in the background on the beach. Look close at the gun Taylor is pointing and you will see the strap hanging free. As the scene returns to a frontal shot you will see the strap is tucked under his left hand again. (01:31:27)
Continuity mistake: When Taylor and Landon discuss Stewart's death on the rocks, their hair goes from being blown in the wind to resting still between shots. (00:13:00)
Continuity mistake: When the 3 of them are swimming and one sees foot prints, he calls to Taylor - when Taylor starts out of the water his hair is wet, but when he stands up out of the water it is dry.
Plot hole: Wouldn't Taylor have suspected he was still on Earth, seeing as all the apes were speaking English?
Continuity mistake: When Taylor ties Doctor Zaius up to the log, shots from behind Zaius show the rope at the top of the log just above Zaius' head to be wrapped around the log only twice. Shots from in front of Zaius show the rope to be wrapped tightly a dozen times around the log. (01:37:35)
Continuity mistake: After Taylor dismounts the second ape, he attacks the ape, and the whip/rope used is dropped and Taylor is a few feet from it. In the closeup, he suddenly has it in hand.
Answer: I think that this is meant to be a mystery. Taylor/Charlton Heston, an astronaut, leaves a world set somewhat in the future after 1968 (when the movie was made) but still recognisable to cinema-goers at the time, to travel through a "time vortex" to arrive in a world in a distant future, which has changed beyond recognition. Taylor meets the orangutan Zaius/Maurice Evans, and Zaius hints that he has some idea of what had happened, but Zaius' knowledge is either limited, or else Zaius is not going to tell Taylor (or his fellow apes) the full story. At the end of the movie Taylor discovers that, at some point between his leaving his own time and arriving in the "Planet Of The Apes", the world had been devastated by a nuclear war, but I think that the exact time, causes of, and course of this nuclear war are deliberately left as a mystery. Sometimes I think a bit of unresolved mystery actually improves a story, and I think this is the case here.
Rob Halliday