fatherhood vs paternity

fatherhood and paternity both are nouns.

Word	Noun	Adjective	Verb	Adverb
fatherhood	Yes	No	No	No
paternity	Yes	No	No	No

As nouns, fatherhood and paternity are synonyms defined as:

fatherhood and paternity: the kinship relation between an offspring and the father

fatherhood (noun)	paternity (noun)
the status of a father	the act of initiating a new idea or theory or writing
the status of a religious leader	the kinship relation between an offspring and the father
the kinship relation between an offspring and the father	the state of being a father