Continuity mistake: When Argyle is sitting in the back of the limousine he hears on the news that terrorists have taking over the Nakatomi building and that their means of communication is through the use of CB radios that they brought with them. Argyle then moves to the front seat of the limousine and turns on the CB which is immediately tuned to that exact channel.
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.
Other mistake: McClane's undershirt is white for the beginning of the movie, but when he emerges from the air conditioning ducts, it is now a dark colored shirt (it looks very dark green) with a lot of stains on it. If this is supposed to represent the same shirt, just very dirty, it could not have come from the A/C duct work, for it was all brand new and the interior shots showed it all to be spotlessly clean. The change in appearance is far too abrupt to be possible.
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