Revealing mistake: When Bishop is doing his knife trick on Hudson it's obviously fast forwarded as you can clearly see the head of Apone in the background move way too fast for real life. No one can move their head that fast without snapping their neck. (00:21:25)
Revealing mistake: When Ripley rescues Newt, an alien comes from a corner and it jumps from side to side. In the second jump, you can see two wires on its back. (01:50:30)
Revealing mistake: When Newt is sliding across the grated floor, Bishop reaches out to save her and you can clearly see he's standing in a hole to make him look cut in half. (02:11:40)
Revealing mistake: When Ripley enters the elevator to go down into the hive, her pulse rifle/flamethrower combo is a stunt prop. The giveaway is the pulse rifle's front is blank (literally just a flat surface with no detail and no holes for the barrel or the grenade launcher). It's particularly obvious in the newer HD versions.
Revealing mistake: When Ripley first burns the egg plants inside the nest, the Queen screams and moves her head. The steel rod used to hold the Queen's head to the rest of her body is easily visible.
Revealing mistake: Right before Ripley discovers all of the eggs that the Alien Queen has laid, look at Newt's fingers as Ripley is carrying her - It's obviously a fake hand. Newt is in fact a dummy in this scene, and others preceding the chase from the Alien Queen. This dummy was created so that Weaver would not get exhausted carrying a real child around for so many scenes/takes.
Revealing mistake: In the Special Edition of the film, in the scene where the Marines are conducting their initial search of the colony, the shot where Hudson and Vasquez enter the "negative situation" room (where they find the hamster) is obviously sped up. You can tell by how quickly Vasquez seems to move when she comes through the door.
Revealing mistake: When the loader is dragged into the airlock by the Queen, during the shot where you see the orange light get smashed on the floor (seconds before it actually falls in), you can see two wires attached to the loader leading off screen.
Revealing mistake: Briefly after landing on the planet, we see the APC trundle across the ground. The way it moves shows it is a model, every little bump makes it jump up and down. This is a shame, considering that when it is inside the facility and in subsequent shots, it looks very good.
Revealing mistake: The "mission time" clocks on the video screens are very inconsistent most of the time they're seen. At several points they jump back and forth from 00:01:31 to 00:01:32 then back again even though all of them are supposed to be synchronized.
Revealing mistake: When the swarm of aliens drops through the ceiling panels, you can tell that Ripley and the marines don't have magazines in their rifles. It's especially visible when Ripley looks up before Hicks raises the panel and sees the aliens.