Anything that happens heavily is weighty, extensive, or labored. A car falling off its jack will crash heavily to the ground, and if you are a new runner, you'll be breathing heavily after the first mile.
Sometimes the adverb heavily simply describes the action of something heavy — like kids' feet pounding heavily up the stairs, or an airplane landing heavily on the runway. There's also the "large amount" or "a lot" meaning, like when rain falls heavily or a director is heavily influenced by the cartoons of her childhood. Breathing heavily, on the other hand, means "with effort."