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).
Suggested correction: This presumes that a person can notice the difference between a fully-loaded pistol versus a pistol that is empty, which makes sense. However, it would be much more difficult for a person to tell that a gun is completely empty versus one that only has one or two bullets loaded. Since Gruber has no way of knowing his ruse didn't work, he doesn't believe McClane would have given him a completely empty gun. He obviously believes there is at the very least one bullet in the gun. He's not planning on getting into a shootout with McClane, he is just planning to threaten him and he certainly doesn't need a fully loaded gun to do that.
Suggested correction: It's easy to tell the different between the weight of a loaded vs. unloaded pistol, but given Hans had never held the pistol before, he had no way to compare the weight. Unless he regularly handled that particular model of pistol (unlikely, given his pistol is a completely different make and model) he would have no point of reference.
Character mistake: When Frank opens the present in his office and finds the picture frame from his brother inside, he reads what is supposedly written on the frame out loud: "To Frank, the best brother a guy ever had. Merry Christmas." However, instead of Merry Christmas, it says Love, James.