Trivia: This film got an amusing shout out at the 2006 Tribute to Bruce Willis. Specifically, the event was hosted by Alan Rickman, who was introduced as 'straight from the 30th floor of the Nakatomi building'. For added humour, Rickman then hobbled onto the stage on crutches and wearing multiple casts and braces as if he was still recovering from the injuries of his fatal drop.
Deliberate mistake: For Hans Gruber's iconic death scene, Alan Rickman's fall was filmed at high speed (for slow-motion playback) against a green screen, and the skyscraper perspective footage was added later as background. However, while Rickman falls away from the camera in slow motion, papers are fluttering around him in the background at normal speed. This was done deliberately to make the shot even more surreal.
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