Revealing mistake: In the scene where Taylor is running after the humans when their clothes are stolen, they're supposedly naked. As Taylor is running through the trees, you can see he is wearing flesh coloured shorts. (00:28:45)
Revealing mistake: As Lucius tackles the guard and yells "Grab him!" to Taylor, Lucius then strikes the guard over the head, knocking him unconscious. When Lucius begins pushing the guard over to Taylor's cell, you can see the guard is already unconscious before he gets struck. (01:18:45)
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.
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.
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.
Revealing mistake: If you look closely at the back of the head of both Zira and Dr. Zaius, you can see the actor's real hair.
Revealing mistake: When Taylor approaches Miss Liberty, there should be horse tracks in the sand extending to the end of the frame. About 1/3 of the frame shows smooth sand preceding some horse tracks.
Revealing mistake: During the trial Honorius walks around the room and stops to deliver his words. Watch his feet and you'll notice a dark spot right where his feet are, which serves as the marker.
Revealing mistake: After Taylor gets up from being dragged in the square, another ape tries to rope him. As this occurs, the stuntman is easily distinguished.
Revealing mistake: During the desert walk you can see footprints in the sand from previous takes, also dark lines in front of them which they follow like a pathway.
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