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)
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: 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 the hostages are running down the stairs after the C4 explosion, a large, fist sized chunk of concrete falls on an older man's head and he just runs straight on. (01:52:35)
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)
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: Even though there is no light shining on John McClane during the fight with Karl, there are several shots where it's obviously not Bruce Willis, but a stunt double.
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)
Revealing mistake: Having leapt from the rooftop, Bruce has his feet planted on the window. As he then goes to push himself back in order to smash the glass with his gun shots watch the rope. Someone above has kindly held the rope out several meters away from the building in the next shot, allowing him to get a much bigger swing.
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)
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.
Revealing mistake: When John McClane jumps from the explosion in the elevator shaft, you can see that it is not Bruce Willis but his stunt double.
Revealing mistake: As Bruce Willis walks into Nakatomi Tower, which is solely occupied by the Nakatomi corporation, behind him can be seen the logo of Merryl Lynch, which had offices in the building used for filming.
Revealing mistake: The first time John walks up to the rooftop, he walks backwards to avoid the thugs' bullets and shelters behind a wall. Despite supposedly being barefoot his socks are very noticeable and wrinkle around the ankle.
Revealing mistake: When John jumps off the roof with the hose tied around him, you can see that the stuntman's hair is slicked back with fireproof gel during the explosion.
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