Deliberate mistake: When the Alien appears behind Brett, we first see its tail lowering before the rest of its body, even though the tail is presumably how the creature was being suspended in the air. Immediately after chomping Brett's head, he and the creature are lifted upwards, again presumably by its tail, though it's not long enough to carry them the implied distance and it's unclear what it has latched onto in the first place. This scene starts a tradition of sorts, as both 'Aliens' and 'Alien 3' feature their own scenes in which an Alien grabs a victim and both are subsequently lifted upwards by a great height, presumably by the creature's tail, and the mechanics by which this is possible and even where exactly they're going off to are always obscured to the point where the creature may as well be wearing a jet pack.
Suggested correction: The alien is suspended from the chains above Brett by its feet and hands. In the Director's cut this is more shown. There are also extended versions of the scene which make it more obvious what happens before and after Brett is bitten. The creature uses the chains to escape to the ducts and those are certainly well within reach of the ground. The scene you refer to in Aliens, in which Dietrich is grabbed, shows the alien using its legs to walk backwards up the wall as it carries her, the tail is not involved. These creatures are shown in all the films to possess an ability to grip walls and other surfaces similar to insects, and they are shown with remarkable flexibility, agility, and strength.

Continuity mistake: Near the start, when Dallas is at the entrance to Mother, one of the lights to his left, the fifth one down from the ceiling, is not lit. The next shot from inside Mother looking out shows that light is now lit. This exact same thing happens a second time when Ripley is entering mother later in the movie. These two scenes must have been shot at the same time. (00:08:20 - 01:18:35)
Suggested correction: That light totally had enough time to light between the frames, because it was hidden by the sliding door for a brief moment. Lights were lit one at the time, not all at once, so this one starts to shine just a little bit later than all the others, that's it.
I consider this a valid mistake and the correction is a stretch. The lights only took about 2 seconds to light up, but she's at the door for almost 3.5 seconds after they're on and the last light did nothing. Then in the half second it took the door to open, the last light is now on.
Continuity mistake: When the ground crew prepares to leave, the captain tells them to take weapons. All three crewman are in suits with side arms on hips. These weapons are never seen the rest of the movie.
Suggested correction: With reference to the side arms mistake the weapons are seen, when Kane goes to look inside the egg before he is attacked he is carrying his side arm and holding it ready.
Trivia: At the premiere of Alien, religious zealots set fire to the model of the alien, believing it to be the work of the devil.
Suggested correction: Myth. While this verbatim description of the vandalism that destroyed Giger's model is still bouncing around the Internet, blaming "religious zealots, " there is apparently no documented factual evidence to substantiate it. In fact, the only source for the "religious zealot" claim comes from David A. McIntee's 2005 book "Beautiful Monsters: The Unofficial and Unauthorized Guide to the Alien and Predator Films, " which has been criticized for too much opinion, bias and speculation, and too little documented fact. In actual documented accounts, the arson of Giger's models at the Egyptian Theatre is blamed simply on "vandals, " with no mention of motivation, religious or otherwise.