family name vs last name

family name and last name both are nouns.

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

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

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

Other synonyms of family name include cognomen, surname.

family name (noun)	last 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)