Deliberate mistake: For Hans Gruber's iconic death scene, Alan Rickman's fall was filmed at high speed (for slow-motion playback) against a green screen, and the skyscraper perspective footage was added later as background. However, while Rickman falls away from the camera in slow motion, papers are fluttering around him in the background at normal speed. This was done deliberately to make the shot even more surreal.
Continuity mistake: When Al gets the call to go to Nakatomi, he's inside the store and walks outside. In the very next shot outside, the door of the store is already closed while Al takes his first few steps away from it. Even if it slammed, it wouldn't close that fast. (00:46:00)
Continuity mistake: After he climbs out of the air duct, John McClane's shirt goes from white to green. The dirt and grime from the air duct would cause discolouration only where the shirt comes in contact with the duct. However, his shirt is uniformly green including places that wouldn't make contact and it is impossible for all surfaces to make contact even in small areas. (00:23:05 - 00:56:30)
Other mistake: Powell says he has a desk job, but he is driving a cruiser. Cruisers are for cops on patrol. A cruiser would not be issued for a cop at a desk, especially not one who's on his way home, as Al says he is.
Visible crew/equipment: In the first scene of the Christmas party when Takagi steps out of one the offices and greets a couple of his guests, the camera pans left to show the orchestra and other guests. A huge shadow of the camera and its operator is being shown on the back wall, then the stage light gets moved to light a different area and the shadow is gone. (00:02:20)
Continuity mistake: When the terrorists launch the rocket at the RV, they break the same window of the building twice. (01:11:55)
Visible crew/equipment: When one of the terrorists enters an abandoned floor, logically the floor should be empty, but you can see the casting crew in the reflection of the windows. (00:35:00)
Continuity mistake: When Hans takes Holly hostage, as she is sitting on the floor talking to Hans, as the camera goes from him to her, each time Holly's shirt is unbuttoned and open more so her bra and boobs are revealed.
Revealing mistake: When Hans discovers the first bad guy lying dead in the lift ("Now I have a gun - HO HO HO!") and pushes his head, the corpse is moving his head before he is hit (watch carefully). He also closes his eyes before Hans makes contact. (00:39:15)
Other mistake: When Bruce is looking on the touch-screen computer to find out where his wife is, he touches the name GENNARO, but when it changes colour it also corrects the spelling of his wife's name to GENNERO. (00:09:25)
Other mistake: In the shot where we see all the terrorists (except Karl and Theo, who are upstairs) walking into the building from downstairs, they are about to walk through a doorway when the camera cuts out. Watch the terrorist on the left (he's the one who guards the door) - the way he's walking he's going to smack straight into the wall! The shot cuts a fraction of a second before he does, but that's not a remotely normal walk/position - it's clearly just because it makes for a cool looking shot. (00:18:15)
Revealing mistake: Bruce Willis wore fake feet during the movie to protect his own while shooting the film. They were like boots that slipped on over his. At the part where the FBI guy shoots at him from the helicopter and Bruce jumps to a lower level of the roof, you can see the tops of the fake feet that aren't tight around his ankles. (01:50:30)
Factual error: The cops try to get in the building by cutting some metal part of the door, it might be the lock. All they have to do is break the glass. SWAT teams aren't going to worry about a bit of property damage with hostages' lives on the line.
Factual error: Throughout the film, characters are able to break through on radio transmissions and speak to the other person talking, and the other person would hear them. In reality, they would have to wait until the first transmitter un-keyed the radio before any other discernable radio traffic could have be heard. (A good example of this is the scene where Ellis is giving up McClane.) (01:18:40)
Continuity mistake: When the terrorists enter the building, they stream out of the rental truck. You don't see the large ambulance that emerges for their getaway at the end of the film. This is because the ambulance was a late addition to the script - the terrorists leaving the truck was originally a longer shot, but had to be cut down as short as possible to hide the lack of ambulance. More info in the trivia section. (00:18:00)
Other mistake: After Powell frantically radios in his distress call, other police cars converge on the street in front of Nakatomi Plaza, blocking off both sides of the street. But when McClane grapples down the building using the firehose, you can see traffic on the street below and none of them are police.
Continuity mistake: When Bruce Willis shoots Alan Rickman, Rickman falls through the window and grabs Bonnie Bedelia's arm to pull her with him. He has a good grasp on her arm because of her watch. When Bruce Willis is trying to release the clasp on her watch, you see blood on her face from his arm. The shot then cuts to Alan Rickman, and when the camera goes back to Bedelia and Willis, there is no longer any blood on her face. (01:57:15)
Other mistake: Near the start of the film, when John goes to meet his wife at the office block, there is a party going on and a small orchestra playing. When Gruber and his gang enter, the orchestra are still there. Gruber locks down the building, and takes the guests hostage. Where does the orchestra go after that? They are never seen again, even when the hostages are taken to the roof.
Revealing mistake: When the hostages are running down the stairs after the C4 explosion, a large, fist sized chunk of concrete falls on an older man's head and he just runs straight on. (01:52:35)
Revealing mistake: When John and the terrorist fall down the stairs together, you can hear the snap of the terrorist's neck signifying his death but at the bottom you can see the terrorist's "corpse" is still breathing. (00:36:00)
Answer: The most likely reason is that the two versions have been "panned and scanned" differently. In the original theater version, both things are on screen at the same time at opposite sides of the screen. In one version, the person who did the TV P&S (not someone associated with the making of the film) chose to move the view from one side of the original picture to the other, showing the terrorist, while the person who did the VHS P&S stayed focused on one side of the frame, only showing Al.
Myridon