Is there any difference if Peeves said “Not doing anything” instead of “Not doing nothing?

Peeves the Poltergeist is a practical joker in Harry Potter books. Why is he saying ‘nothing’ for ‘anything’ in the following citation? Does this express his character? Is it possibly a dialect or anything?

”Peeves, get down here NOW!” barked Professor McGonagall, straightening her pointed hat and glaring upward through her square-rimmed spectacles.

”Not doing nothing!” cackled Peeves, lobbing a water bomb at several fifth-year girls, who screamed and dived into the Great Hall.

N.B.: Peeves is floating in the air and throwing water bombs.

"Not doing nothing", is not, as some have asserted, grammatically incorrect. It is just pretty likely that the speaker means something different than the strictest interpretation of the phrase would mean.

For example, if I am talking to my kid:

Me: "Kid, it is a beautiful sunny day, how come you are sitting here doing nothing?"

Kid: "I'm not doing nothing; I am imagining what it would be like to travel to Jupiter."

Here, the kid is using the phrase correctly. However, most likely Peeves does not have that intent at all, he really means "I am doing nothing." It additionally has a color of sulky and sophomoric. It is one of those phrases that people frequently misuse, and most likely Ms. Rowling is trying to convey a particular flavor of the pesky poltergeist's personality.

It is fair to ask if it is actually incorrect. If many, many people use a particular phrase, and its meaning is well understood, then, the goosebumps of strict grammarians notwithstanding, it might be argued that it is at least colloquially correct. Frankly, I could care less, or I couldn't care less, or something like that :-)

Peeves is using negative concord, where two negatives reinforce each other. This is a feature of some dialects.

Yeah, it is grammatically incorrect, and was intentionally written so to indicate Peeves' personality/background.