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).
Factual error: A few of Gruber's henchmen set up the rocket-launcher to blast the armored police vehicle. As one of them swings the legs of the launcher down, another uses a 'Hilti' gun to anchor the leg in place. A 'Hilti' gun is a construction device that uses gunpowder (usually a .22 shot) to 'blast' a nail into very hard material such as concrete. With the model they're using, the nail has to be manually loaded into the front of the gun for each shot before it can be used, however the guy using the gun never loads a nail into the gun. As it is, the result would be worthless.
Stupidity: Hans keeps a major part of his plan secret from his own team: that the electromagnetic lock will be disabled if the FBI shuts down power to the building. The mercenaries hired as muscle don't need to know the minutiae of the plan, but it seems ludicrous that Theo wasn't told. Theo states on more than one occasion that he can't proceed past a certain point and that he hopes Hans has a plan for the final lock. Evidently, Hans was keeping this information secret simply to amuse himself, which makes little sense considering how much planning went into the heist.