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 the little boy is reading his report about Abraham Lincoln, he states that Lincoln was born in a log cabin in Illinois. Lincoln was actually born in a log cabin in Kentucky.
Character mistake: When Mick arrives at walkabout hotel, Nugget introduces himself to Mick as if they have never met before. Yet in the first movie, Nugget is asking how Mick is getting on in New York. It may be just a case of over familiarity by Nugget but it doesn't make a lot of sense.
Suggested correction: It's quite possible that they haven't met, but Nugget's heard about Mick, e.g. from the people in the bar, stories, etc. Having no doubt heard so much about Mick, Nugget probably feels like he knows him, at least enough to ask how he's getting on.
Character mistake: Mac's cousin says that even if Nick wasn't a cop, he'd have to hate Mac's guts. He says that Mac wears a Rolex and Nick wears a Timex - not true. Nick is wearing a Rolex, also.
Character mistake: When Nico comes out of the parking lot and gets caught by the police, right before he gets in the van and is patted down by the police officer, the police officer says, "she's clean", not "he's clean."
Character mistake: When Dr. Walker and Michelle are dancing at the club, the people around can't help but stare at the camera.