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: 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)
Factual error: Bruce Willis arrives in Los Angeles on Christmas Eve around 6 p.m. There is no way we would see that beautiful golden sunset we see. During that time of the year, the sun would have set more than an hour earlier. (00:07:50)
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)
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)
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: 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)
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)
Continuity mistake: When Takagi explains to the terrorists that, "he doesn't have that code," the trees outside the window are clearly swaying in the wind. The next shot is a panoramic view of the room, but now, the trees outside are still. (00:27:50)
Visible crew/equipment: When McClane is crawling under the long table to get closer to hear the terrorists' conversation, you can see the reflection of the camera operator's legs following along with him on the table base. (00:29:15)
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: 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)
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)
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)
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)
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)
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)
Visible crew/equipment: In scene on the top floor after John hits the fire alarm, as he is seeing the fire trucks coming and then turns away, once the lights are turned on, you can see the reflection of a number of crew people in the window during/after the cut away. One dude in a green shirt, another in a white shirt are prominent. (00:41:00)
Continuity mistake: When he's climbing down the vents, the strap unties, and Bruce Willis falls. He misses the vent he was reaching for, and falls past the next one, so we assume he catches the third one down. However, after he flicks off his lighter, the camera pans up from the lit vent to Karl, and we can see that he somehow grabbed and climbed into the vent he definitely fell right past in the shot from above. (00:48:30)
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)
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