Visible crew/equipment: In the first scene of the Christmas party when Takagi steps out of one the offices and greets a couple of his guests, the camera pans left to show the orchestra and other guests. A huge shadow of the camera and its operator is being shown on the back wall, then the stage light gets moved to light a different area and the shadow is gone. (00:02:20)
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)
Visible crew/equipment: When McClane is crawling under the long table to get closer to hear the terrorists' conversation, you can see the reflection of the camera operator's legs following along with him on the table base. (00:29:15)
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)
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)
Visible crew/equipment: When Al goes into the lobby of the building, he walks over to a large reflective marble counter. If you look at this counter you can very plainly see the reflection of a large metal stage lamp hanging right over the actors' heads. (00:52:25)
Visible crew/equipment: Everyone is surprised when Karl is alive at the end of the movie because he appeared to be dead when hanging by the chain that McClane had wrapped around his neck. However, if you look at the back of Karl's head during the entire scene, you can see the safety wire coming out of the back of his shirt that holds him up. No wonder he survived. (01:48:45)
Visible crew/equipment: Just after John has shot Hans, he falls and smashes the glass window. Look behind him: you can see a wire in two shots. It's meant to hold the stunt double for either Holly or Hans. No slow-mo required.
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.
Visible crew/equipment: At the end when all of the papers are falling from the building, there is a wide shot. On the right hand side of the shot looks to be some kind of blower machine making the papers come out.
Visible crew/equipment: McClane runs up the stairs and looks on floor 31. He goes up another floor as the camera follows. The camera operator's shadow is blatantly visible left screen.
Visible crew/equipment: As the terrorists arrive, there is a shot of two vehicles pulling up. The camera is tracking along with the vehicles. At the very end of the shot its shadow and a crewmember's shadow are visible on the concrete.
Visible crew/equipment: When McClane steps out of the Limo at Nakatomi, a large film light is reflected in the door.
Visible crew/equipment: When the car and truck drive upto the building, the truck goes down a ramp - another moving shadow can be seen cast over the car, from the camera crew.
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