cognomen vs family name

cognomen and family name both are nouns.

Word	Noun	Adjective	Verb	Adverb
cognomen	Yes	No	No	No
family name	Yes	No	No	No

As nouns, cognomen and family name are synonyms defined as:

cognomen and family name: the name used to identify the members of a family (as distinguished from each member's given name)

Other synonyms of cognomen include last name, surname.

cognomen (noun)	family name (noun)
the name used to identify the members of a family (as distinguished from each member's given name)	the name used to identify the members of a family (as distinguished from each member's given name)
a familiar name for a person (often a shortened version of a person's given name)