When Dr. Hasslein opens the trunk of his car to retrieve the binoculars, he is seen removing the keys to the trunk from his pocket (this in itself is an error as he just parked; we're to believe he put the keys in his pocket whilst still seated in the car). In a subsequent shot where he opens the trunk again (to retrieve his revolver), he retrieves no keys from his pocket and the trunk key is already in the trunk lock and clearly visible, having been there when he parked.
The ship the apes arrive in has no engines or tail section as it did in the first film (it was under water) or Brent's in the second one (it was burnt up).This is because the ships were designed to land on their destination planet, then lift off again. Once back in Earth orbit, the rear section is automatically jettisoned before re-entry.
(at around 27 mins) Whilst in captivity, Cornelius explains that Dr. Milo salvaged Taylor's space ship (which crashed and then sank in the first film) and rebuilt it, which allowed the three apes to escape from the Earth before it was destroyed and travel backwards in time. The apes in the 40th century did not have the technology to salvage the ship from the bottom of the ocean, let alone the knowledge to rebuild it and learn how to fly it.