family name vs surname

family name

noun
  • A given name common in a family, or already borne by someone in the family (in the context of naming someone in the family after that family member). 

  • A surname. 

surname

noun
  • The name a person shares with other members of that person's family, distinguished from that person's given name or names; a family name. 

  • The cognomen of Roman names. 

verb
  • To call by a surname. 

  • To give a surname to. 

How often have the words family name and surname occurred in a corpus of books? (source: Google Ngram Viewer )