Factual error: When we see pictures from the cockpit of the planes we can see the individual snowflakes. That is impossible with the speed an airplane is flying.
Other mistake: McClane pulls out a 9mm Heckler and Koch MP5 submachine gun, aims it at the police chief and pulls the trigger. It's loaded with blanks but nobody except him knows that. The problem is that the police chief is surrounded by armed colleagues, and not one of them draws his gun in defence of his boss. In reality McClane would be riddled with bullets before he could pull that trigger. Not to mention an MP5 needs a blank firing adapter to use blanks (it fits on the muzzle via three lugs), none is fitted on the gun in the scene. As a blank firing prop gun it would use a hidden internal restrictor, but that wouldn't be feasible if it was meant to be able to switch between blanks and regular ammunition, as it's supposed to in the movie.
Plot hole: There is no way the airport personnel could have kept everything going on quiet from the thousands of civilians in the airport. Even if nobody heard the shootout between McClane and the terrorists at the baggage area someone would have found out about the crashing of the Windsor flight.
Audio problem: When the terrorist at the church goes inside and asks "where's Cochrane", the other replies "he didn't make it." His mouth isn't moving.
Continuity mistake: When McClane lights the torches on the runway he is surrounded by rough snow, then the long shot shows a smooth runway in the snow, then when the plane crashes the pilots see the tarmac with no snow, then the outside shot of the plane crashing shows the runway covered with snow again.
Revealing mistake: When the army guy opens the door on the 747 to climb out onto the wing, the outside door handle does not move and looks painted on. When the second guy re-enters the plane and shuts the same door, there is no locking mechanism shown on the sides of the door; it simply pushes open.
Audio problem: When General Esperanza is trying to land the plane there's a hell of a lot of noise from the wind blasting in through the shattered windscreen. Yet when we see McClane listening in to his radio communication he can be heard completely without a trace of wind noise.
Factual error: There is absolutely no way that Bruce Willis could have stood in the middle of an airport landing zone trying to keep the plane from crashing into the ground. He has no protective ear equipment on - that noise most definitely would have blown his eardrums out and blew him several yards away physically when it flew over him.
Continuity mistake: In the beginning of the film when two of Stuart's men arrive at the church, their van stops and it is not snowing outside,same thing when they're shown knocking and then when the man opens the door. However, in the shot just after (which is taken from behind the door) snow is falling on them. The shot then switches to an outside view ,and it's not snowing anymore .
Continuity mistake: When the army truck arrives at the 747 hangar, we see a line of snow on the ground behind it. The actual truck has no snow on it at all, despite its journey to the hangar.
Factual error: The first shot of the church is a high up shot which pans down as the van enters the shot. High mountains can be seen in the background. The film is set in Washington D.C. (Note: the film was shot in the Denver area.).
Factual error: There is no Glock 7, and Glocks are produced in Austria, not Germany.
Revealing mistake: When the terrorists cut the pipes with wires behind the church, one guy uses a chainsaw, and the pipes he cuts are all empty.
Continuity mistake: While Barnes is brainstorming ideas with his staff about using the marker beacons to communicate with the circling planes, he has a pen in his right hand held to his face. The next shot he turns, but the pen is no longer in his hand and his hand is nowhere near his face.
Audio problem: When McClane goes to Lorenzo's office for the first time, we hear Lorenzo say "who is it? Come in." His lips don't move (which you can tell despite his mug being up to his mouth).
Continuity mistake: When the lady sitting next to Holly pulls out her stun gun, her book is resting against her stomach. However, in the previous shot from a different angle, the book isn't seen.
Visible crew/equipment: When we first go to the Annex Skywalk, the SWAT team are walking with the guy from the tower and he is talking on a radio. In the mirror behind them you can see the boom mike and operator as they walk.
Factual error: In the tower, Leslie Barnes and a few men are talking about the Annex Skywalk, mentioning that it is a "VHF System." Barnes says the planes are so close it won't matter. VHF, or Very High Frequency, transmits far distances, hence why it is used by airports around the world. He should have said "UHF", which transmits shorter distances, as that would make more sense.
Continuity mistake: In the airport lobby, when the two policemen come at the bar they have "snow" all over their clothes. In the next shot they are shown from behind sitting at the bar and Lorenzo's brother has no snow on his jacket. Seconds after Mc Clane comes to talk to them and the cop has snow on his shoulders again.
Factual error: A dangerous prisoner like Esperanza would not be flown into a civilian airport. He would be transported to the nearest USAF base, in this case Andrews Air Force base, always assuming that he is being extradited to Washington D.C.
Suggested correction: Snow doesn't generally accumulate on a moving vehicle. Anything resting on the vehicle will blow off once it's in motion unless it's very wet and/or heavy.
As a truck driver, I can tell you that the conditions shown would have snow accumulating on a moving vehicle at least in some places.