surname vs family name

surname and family name both are nouns.

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

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

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

Other synonyms of surname include cognomen, last name.

surname (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)