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)
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)
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)
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)
Trivia: For the shot where Hans Gruber falls from the top of the building, Alan Rickman was dropped by a stuntman from a 20-foot high model onto an air bag. To get the right reaction, the stunt man dropped Rickman on the count of two instead of three. Rickman said "That sounded like fun and so I decided to do it, I thought they were a bit surprised that I said I would and I was surrounded by people saying, 'You're mad, you're mad!' But no, it was fun. And it was the last shot, so if I'd broken my neck, they'd still have a movie."
Continuity mistake: Takagi is sitting on the chair facing the door of the room when he is shot. In the next scene, the same door is splattered with blood, yet the glass structure which should be splattered with blood should be the glass window behind Takagi, not the door which he is facing. (00:30:15)
Continuity mistake: At the start in the limo, when Argyle is listing all the features of the car, the bear is dead centre of the back seats. The very next shot is in the rear view mirror, and the bear's now over by the window. (00:06:00)
Revealing mistake: John McClane stands in a little pool when the helicopter falling from the roof explodes outside and the windows shatter. In a short shot you see that this man is definitely not Bruce Willis. (01:52:50)
Visible crew/equipment: When Al goes into the lobby of the building, he walks over to a large reflective marble counter. If you look at this counter you can very plainly see the reflection of a large metal stage lamp hanging right over the actors' heads. (00:52:25)
Revealing mistake: When John fights against the blonde German thug, Bruce Willis' stunt and the thug's are very noticeable: different body build plus darker and longer hair in the case of Willis, and bulky wig in the case of the German. (00:35:21)
Continuity mistake: When Powell is driving his cruiser in reverse to get away from the bad guys shooting at him, several shots show a full roll-cage in the car which is not there in any other scene or close-up of the interior. (00:55:10)
Factual error: When Willis kills the terrorist and throws him out the window onto the police car, the body lands on the windshield, and it shatters into little bits. The windshields on 80's cars were double paned with a thin, strong plastic between each pane, to keep them from breaking into little bits in accidents. Only the side and back windows would shatter like that. (00:54:45)
Revealing mistake: After escaping from the roof, John walks inside the building. When he runs towards a huge vent his bare feet swap to wearing black shoes.
Revealing mistake: After barefoot John rolls down the stairs with the German thug one can notice the top of John's (his stunt in this case) protective sock around his left foot. (00:35:44)
Continuity mistake: When Gruber is given a tour of the closed vault, Theo casts a shadow on the wall which changes position when he says the 7th lock's circuits cannot be cut locally. (00:32:00)
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)
Visible crew/equipment: When John gets out of the limo at Nakatomi Plaza, a stage light is reflected clear as day in his door. (00:08:10)
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