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 the terrorists launch the rocket at the RV, they break the same window of the building twice. (01:11:55)
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.
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)
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)
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)
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 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)
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)
Continuity mistake: When the terrorist Karl is about to kill the man in the front desk, his fringe changes from neat and combed to slightly messy from one frame to the other.
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)
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)
Continuity mistake: When McClane has killed the first terrorist and sends him down the lift, McClane is hidden in the lift shaft. During the ensuing conversation McClane writes the names "Hans" and "Karl" on his wrist in permanent marker. This disappears in the next scene.
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)
Continuity mistake: The scene after John jumps from the roof and enters the building, he jumps in a pond of water to evade the helicopter blast and comes out completely drenched, but in the next scene he is completely dry.
Continuity mistake: After John has crawled through the vents and enters the room where Hans shot Takagi, the amount of blood on the window is a lot more than was thereafter Takagi got shot.
Continuity mistake: The headlights on Al's cop car are off in McClane's view from above but back on from ground level.
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)
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