(at around 23 mins) In the fight in the gym, Ripley hits Christie with the basketball, knocking him to the floor. As he falls, he is no longer holding the barbell, but there is no sound of it hitting the floor.
(at around 1h 27 mins) When Ripley awakens in the Alien nest, the shot opens with one of the scientists speaking about evolution. Despite the speech delivered in his voice, the actor seems to just be making random movements with his mouth, as if he is literally saying "Blah blah blah".
The look of the legs does not correspond between the practical and computer-generated Aliens; the CG ones have an extra joint at the ankle, while the practical Alien shot by Vriess is shown with human looking ankles (i.e. without an extra joint) as it is seen crouched on the grate from below. Also, it lacks the long toes of the CG Aliens, instead having more human-like feet.
In some scenes just a few drops of Ripley's blood burns through thick metal. When Ripley undergoes surgery, the alien is removed with forceps and in a deleted scene Gediman holds open her chest with spreaders, her blood should also burn the surgical tools.
(at around 5 mins) After the operation, Ripley is held in a "staging area" in what looks like a thin sleeping bag. In the first scene her legs and her head clearly stick out. In the next scene, she is inside the bag and has to slice it open with her fingernails.
Near the end of the film, Ripley flicks blood from the cut on her hand onto the window of the ship, which burns it. Just seconds later, when the room decompresses, she grabs a chain with the same hand and it doesn't get burned.
(at around 1h 24 mins) After Ripley falls into the Queen's nest, a pan out shot shows the Auriga heading towards Earth, however, as seen by the visible landmasses, the image is transposed (East Africa, Red Sea, and Arabian Peninsula have been flipped across a vertical axis).
With the release of this film, the Alien Trilogy VHS set became the Alien Saga. However on DVD, it was renamed to the Alien "Quadrilogy". Quadrilogy is not a real word. The legitimate word for a series of four is "Tetralogy".
In the first three films, it takes an extremely long time (long enough to require freezing of the passengers) to get between populated places in space. In this film, the ship is able to travel from "unregulated space" to Earth itself in as little as three hours. However, we have no idea what "unregulated space" is (it could be very near Earth for all we know) and there have been two hundred years of technological development since the first three films anyway.
(at around 30 mins) When Ripley drives Call's knife through her wrist it does not get damaged by the acid in her blood even though it's soaked in it. In other scenes just a few drops of Ripley's blood burn through thick metal, which would suggest that the knife's blade should also get burnt. However, Call's knife may be made by some alloy, not just simple steel, but added with cadmium/vanadium/iridium that make it unassailable by acids.
(at around 44 mins) When the alien reaches up through the grating and they show Elgyn getting pulled down, there is a big hole burned into the grate in front of him that was not there just a second before.
(at around 1h 35 mins) Near the end just before when the newborn is sucked out into space, the hole in the window is just too small to create a suction of that strength as seen in the film, and one atmosphere of air pressure would be too little to squeeze the alien out of the ship regardless.
(at around 41 mins) Most of the region of the brain where General Perez is wounded is responsible for processing visual information. The physical shock alone of such a blow, even discounting the destroyed tissue, would leave him blinded and unable to look at anything.
Near the end of the film after the human/alien hybrid had just been sucked out of the hull breach, the ship is shown to be entering Earth's atmosphere. Even though the entire ship is awash with flames reaching 3000 degrees, the only effect in the room with the breach is sparks and flame coming from the hole. In reality, the interior of the hull would have begun to melt, and the increase in drag from the breach would have thrown the ship wildly off course, or even into a spin.
When Ripley's blood breaches the window in the Betty's cargo hold and the Newborn is sucked out, judging by size of the ship, a cargo hold of that size should've been decompressed in mere seconds. The time it takes for the ship to fully enter Earth's atmosphere would've been more than enough to decompress the entire ship, let alone the cargo hold.
If Ripley's gift to the Alien Queen is a human reproductive system, why is she laying eggs? As a human female develops one ovum an a cycle, the Queen Alien does herself; the Eggs were the first cycle, the Newborn hybrid was the second.
After Ripley falls into the Alien's Den and taken to the Queen, that is when she finds out her gift to the alien creature is a human reproductive system. The Queen gives birth to the new hybrid. However, if that alien was given a human reproductive system, how was the queen impregnated? The Aliens reproduce in an asexual way. The Queen fertilized herself and gave birth to the hybrid.