Character mistake: An unloaded automatic pistol is much lighter and balanced differently than a loaded one. When McClane hands "Bill Clay" (Gruber) the empty pistol, a terrorist of Gruber's credentials should pick up on the odd balance (I notice that difference, and I very rarely even pick up a gun).
Character mistake: After the police scanner broadcasts the shooting at Nakatomi Plaza, reporter Richard Thornburg starts begging the station producer/director for a van and cameraman. This is as the evening news is about to go on the air. The floor manager is counting down, "We are on the air in, four, three, two, one...." as he is giving the hand signals to the news anchors. On a real television set, the verbal count would end at three and the mikes would go live. In this case, the floor managers "two, one" and all other words spoken after "three" would be heard by the "at home" audience. (00:58:45)
Character mistake: The psychologist on the TV news is talking about the "Helsinki syndrome", the news anchor comments that Helsinki is in Sweden (before being corrected). There is no such thing as a Helsinki syndrome. However there is a thing called the "Stockholm syndrome". This refers to an actual hostage situation in 1973 in Stockholm, Sweden, when bank robbers held the staff in a bank hostage for several days. During the time in the vault, the hostage and the bank robbers developed a strange kind of friendship, and this is called "the Stockholm syndrome".