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.
Continuity mistake: At the beach when Cornelius reads the scroll his fingers swap between being under the scrolls to being on top.
Visible crew/equipment: After Taylor exits the museum he is whipped by an ape on a horse. Taylor/Charlton Heston is replaced by a noticeable stunt. (00:58:57)
Continuity mistake: When Taylor discovers Stewart's corpse his hand is on his chest but a shot later it's by his side.
Audio problem: After Taylor's been hit Dr. Galen says "he's lost a lot of blood", but his mouth is not moving.
Continuity mistake: While during the trial Honorius says Zira and Galen experimented on Taylor his right-hand swaps between being by his side or resting on the chair.
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.
Continuity mistake: When Julius beats Taylor he protects himself. The position of his hand changes between shots.
Visible crew/equipment: When Zira locks Nova in Taylor's cell watch the left iron bar and you'll see the shadow of the camera approaching.
Continuity mistake: Zira visits the cages and asks Taylor if it still hurts. Taylor holds the iron bars with both hands parallel to each other. A shot later his right hand is positioned higher.
Continuity mistake: When the three arrive at the planet Taylor asks for water and Dodge shows his canteen. Check the strap of his backpack changing between shots.
Continuity mistake: When the movie begins Taylor starts rolling up his left sleeve with his right hand. Shot changes and his hand is away and the sleeve is already rolled. Up.
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