morality vs rightness

morality and rightness both are nouns.

Word	Noun	Adjective	Verb	Adverb
morality	Yes	No	No	No
rightness	Yes	No	No	No

As nouns, rightness is a hyponym of morality; that is, rightness is a word with a more specific, narrower meaning than morality:

morality: concern with the distinction between good and evil or right and wrong; right or good conduct
rightness: according with conscience or morality

Other hyponyms of morality include righteousness, conscience, good, goodness, chastity, sexual morality, virtue, ethicality.

morality (noun)	rightness (noun)
concern with the distinction between good and evil or right and wrong; right or good conduct	according with conscience or morality
motivation based on ideas of right and wrong	conformity to fact or truth
	appropriate conduct; doing the right thing
	conformity with some esthetic standard of correctness or propriety