Hello Ant ?!? ;) Both have the same meaning regardless from the presence or absence of that friend , but "my friend" is more definitive while " friend of mine " usually " a friend of mine " is more indefinitive.

"my friend" - When your friend is there with you "friend of mine" - when you are talking about your friend, but your friend isn't there with you that's the way it's commonly used, but you can use them either way. I think those two are a formal and informal way but not sure which is which.