Well in English, they mean the same thing. However, these days, to avoid confusion, it is common to refer to the surname as 'the family name", and your other names as "given names". That way, it doesn't matter whether in your culture, you write your family name, or your given name/s last.

In the U.S., we rarely use "surname" or "given name". We prefer "last name"" and "first name."

We use first, middle and last name in the States.