Thought about it while reading this question, specifically the quote in the answer:
He bent over her like a cloud, and his eyes glittered; he raised his mace to kill.
In the books, the Witch-King has a mace when he fights Éowyn. In the movies, he definitely has a flail and sword combo.
My initial thought was that Peter Jackson wanted to avoid too many similarities between Sauron in the intro to the first movie and the Witch-King in the third. Helmet aside, they'd look pretty similar if they had essentially the same weapon. But there are other characters with maces, too, notably Azog the Defiler, a character Peter Jackson himself invented, so clearly he doesn't seem to mind an enemy having a mace, unless they have a scary helmet, maybe?
Is there any official statement from Peter Jackson or another important LOTR figure explaining why the Witch-King had different weapons? Or is this just another of those "Hey, it's different from the books for no reason"?