This one's easy. If someone is maniacal then they're behaving like a maniac. Add an -ly to make it an adverb, and you get maniacally, as in "she's acting maniacally."
The noun maniac is almost always used to describe people who do nutty things—serial killers, insane people on the street, crowds at soccer matches. But the adjective maniacal can be more loosely applied to situations, settings, animals, etc. You can use maniacal to describe almost anything that seems wildly crazy or dangerously out of kilter. I don't even remember what the Beatles played that first time on TV. It's the maniacal screaming from the fans that sticks in my head. No movie villain was ever as maniacal as Jason, in Friday the 13th.