It's the same thing in English. But not always in other languages, which can cause confusion...

In English, they mean the same. We typically only have one last name, unlike Spanish or Portuguese-speaking cultures. There, it's not uncommon to have four names. First, second, dad's last name, or mom's maiden name. I'm in Guatemala. Here, this is the standard convention.

Its the same thing. Your family name is your last name and it is often referred to as the surname.

In US English they are the same, however „last name“ sounds a bit more colloquial to me, and is most likely used far more frequently. Family name sounds a little foreign to me, to be honest. Other languages use that word/phrase. Even on official US government forms you’ll see „last name“ printed rather than „family name.“ It‘s a bit like saying „given name“ rather than „first name.“ I’d use „first name.“ I’ve very rarely heard anyone say or write „given name.“