distressfulness vs badness

distressfulness and badness both are nouns.

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

As nouns, badness is a hypernym of distressfulness; that is, badness is a word with a broader meaning than distressfulness:

Other hypernyms of distressfulness include severeness, severity.

distressfulness (noun)	badness (noun)
the quality of arousing fear or distress	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