Character mistake: During the St. Ives riddle, Simon gives them 30 seconds to solve the riddle and call back. But it takes them 65 seconds to call, to which Simon says they're 10 seconds too late. And, after 42 seconds, McClane says they only have 10 seconds left.
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.
Chosen answer: A simple omission, in the heat of the moment he forgot to mention it. He may have thought, since he just took the gun off an enemy, that the safety was already off.