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.
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.
Plot hole: After they shoot the glass and McClane escapes, the terrorists all just leave. There are limited ways out of the room, there must be a blood trail over the glass, he's moving slowly...and they let him go.
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)
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)
Factual error: John McClane wraps a fire hose around his waist, uses it to jump 100 feet until the coil catches the lip of the building, and the hose catches him. This should have seriously injured him due to the inelastic material fire hoses are made of, and the fact that he was not wearing a safety harness, but he barely gets the wind knocked out of him.
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.
Other mistake: When John is on the roof being shot at, he shoots the lock off a door before charging through it. We see a close up of the door as it's shot, and there's a blood smear already on it. He then goes through it shoulder first, which is what would leave a smear like that - must be from an earlier take.
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.
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)
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)
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)
Factual error: If the hose hanger isn't heavy enough to instantly yank McClane out of the window, but instead slowly pulls him down, then it should be light enough for McClane to simply stand up and reel it in. If his weight is enough to slow the hanger's decent, then that means it only weighs slightly more than he does. He should have no trouble standing up and walking backwards to pull the hanger back up. The mistake is that the hanger pulls him slowly at all, it should have yanked him directly out of the window the second the line ran out of slack because McClane is not anchored to the floor in any way. This scene suggest that the hanger is both heavy enough to pull McClane but light enough that McClane's weight significantly slows it, which is impossible.
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)
Continuity mistake: When the FBI shows up and Duane Robinson is explaining to them what the police know, Al tells the FBI about John. Duane is in front of Al in the shot looking at the building, and right next to him in the opposite angle.
Visible crew/equipment: The shadow of the camera is visible right of screen when John is wrapping himself with the fire hose and asking himself why he's doing it. It lingers for some time. It looks to be a steadycam with a monitor.
Continuity mistake: Hans leaves his gun and flashlight lying on the ledge out of sight. When he edges back to get it, the gun is upright, easy to grab.
Revealing mistake: When McClane is on the roof shooting over the hostages the helicopter appears and they start shooting at McClane, the squibs used for the helicopter shooting scene are visible lying on the roof and explode in sequence, starting at the back and moving toward the camera as McClane does.
Continuity mistake: When McClane goes onto the roof at the end he shoots Uli as he opens the door and the blood splatters all over the wall at the side, but when McClane walks past the body there is no blood on the walls or steps or where the body is lying.
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