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)
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)
Continuity mistake: When Powell's police cruiser is first fired on, he accelerates in reverse quickly. The first obstacle he hits completely destroys the right-hand side tail light assembly. Subsequnt shots of the car show the tail-light to be totally intact again. (00:55:05)
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: The news reporter is listening to the police scanner, when Willis is heard talking, the channel lights that blink on the scanner are supposed to stop on the channel being used, but they keep flashing. (00:55:50)
Character mistake: After the police scanner broadcasts the shooting at Nakatomi Plaza, reporter Richard Thornburg starts begging the station producer/director for a van and cameraman. This is as the evening news is about to go on the air. The floor manager is counting down, "We are on the air in, four, three, two, one...." as he is giving the hand signals to the news anchors. On a real television set, the verbal count would end at three and the mikes would go live. In this case, the floor managers "two, one" and all other words spoken after "three" would be heard by the "at home" audience. (00:58:45)
Continuity mistake: In most of the film Bruce Willis' shoulder scar (from surgery when he was 17) is on his left, but as he is looking out the window just before SWAT comes in, it's on his right, clearly a flipped shot. (01:11:40)
Continuity mistake: When the terrorists launch the rocket at the RV, they break the same window of the building twice. (01:11:55)
Continuity mistake: In the scene where Al asks John if he has any kids of his own, John pulls out his wallet with a picture of his family. He is holding the radio and the wallet, but when it cuts to John's "Jungle Gym" comment, both his hands are on the wallet. It then cuts to John again holding the radio and the wallet. (01:16:35)
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)
Audio problem: Just after Hans addresses McClane by his name over the radio, we can hear Powell say, "Better get ahold of someone at dispatch," but his mouth movements don't match what we hear. (01:18:50)
Factual error: The shrink on the news refers to the Helsinki syndrome. This should be the Stockholm syndrome. Being an expert, he should know the difference, and there's no suggestion it's a deliberate oversight to imply a lack of knowledge. (01:25:25)
Character mistake: In the infamous 'shoot the glass' scene, Alan Rickman, playing a German, commands to 'Schieß den Fenster', which is wrong in German because the 'Fenster', the window, has in fact the neutral article 'das' instead of the masculine 'den'. Also, it is not a window in German but a 'Fensterscheibe', a window pane. In German he would say 'Schieß auf das Fenster' ('Shoot at the window!') or 'Schieß auf die Scheiben' ('Shoot the window panes!'). (01:36:00 - 01:37:30)
Continuity mistake: When John and Karl are fighting and Karl reaches for the gun lying on the floor, John legs it out of an open door. Karl fires two shots, the first hits John and the second sprays blood all over the door. (01:47:35)
Visible crew/equipment: Everyone is surprised when Karl is alive at the end of the movie because he appeared to be dead when hanging by the chain that McClane had wrapped around his neck. However, if you look at the back of Karl's head during the entire scene, you can see the safety wire coming out of the back of his shirt that holds him up. No wonder he survived. (01:48:45)
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)
Revealing mistake: Bruce Willis jumps off the roof with the firehose. The hose hanger breaks off the wall and apparently gets stuck in the railings. The shot is very careless: slow motion shows that the firehose doesn't get stuck, but simply clings like a magnet to the top of the railings. (01:51:50)
Continuity mistake: After John McClain has jumped off the top of the building, and the firehose hanger has got stuck to the railing, he has stopped at the top of the window, but in the next shot he is near the middle. (01:51:55)
Continuity mistake: After the terrorists blow the roof and Willis jumps off with the fire hose, he hits the window. After kicking the window for a while, he decides to shoot it. As he pushes off the window you can see that his gun is out and pointed at the window yet in the next scene he pulls the gun from his side and then starts shooting at the window. (01:52:00)
Continuity mistake: When Bruce Willis breaks through the window (as the roof is blown), he lands on his back, head pointing towards the window. However the next moment we see him catching his breath, lying on his stomach and feet aimed at the window as the firehose hanger falls past the window. (01:52:10)
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