naughtiness vs badness

naughtiness and badness both are nouns.

Word	Noun	Adjective	Verb	Adverb
naughtiness	Yes	No	No	No
badness	Yes	No	No	No

As nouns, naughtiness and badness are synonyms defined as:

Other synonyms of naughtiness include mischievousness.

naughtiness (noun)	badness (noun)
an attribute of mischievous children	that which is below standard or expectations as of ethics or decency
	used of the degree of something undesirable e.g. pain or weather
	an attribute of mischievous children