father vs pater

father and pater both are nouns.

Word	Noun	Adjective	Verb	Adverb
father	Yes	No	Yes	No
pater	Yes	No	No	No

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

father: a male parent (also used as a term of address to your father)
pater: an informal use of the Latin word for father; sometimes used by British schoolboys or used facetiously

Other hyponyms of father include dad, dada, daddy, pa, papa, pappa, pop, father-in-law, old man.

father (noun)	pater (noun)
a male parent (also used as a term of address to your father)	an informal use of the Latin word for father; sometimes used by British schoolboys or used facetiously
the founder of a family
a person who holds an important or distinguished position in some organization
a person who founds or establishes some institution
the head of an organized crime family

father (verb)	pater (verb)
make (offspring) by reproduction