last name vs family name

last name and family name both are nouns.

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

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

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

Other synonyms of last name include cognomen, surname.

last name (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)