Revealing mistake: After McClane chases the ambulance in the cab, he is driving the wrong way down a one way street. Notice the tyre marks on the pavement from previous takes. The cab follows them to a T.
Revealing mistake: When the bomb explodes at Bonwit Tellers at the start, we see a van fly into the air. You can see the devices that throw it upwards.
Character mistake: When McClane and Zeus are in the cab travelling east on 72nd street, as they approach the park Zeus says "we need to go south!" he points to his right, which is north. When McClane crosses 5th Ave., he turns his steering wheel to the right, thus heading north.
Answer: And Zeus mentions about brothers knowing how to shoot guns.
He said it was racist to assume that brothers know how to shoot guns. He admitted he didn't know how to use that model.