family name vs surname

family name and surname both are nouns.

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

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

family name and surname: 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, last name.

family name (noun)	surname (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)