Haha guys, really I promise you there is no difference between them, neither one supplies more information at all (see below), I am from the UK and I speak the queen's english so I can assure you either is perfectly fine, I'm sure if you ask any native speaker from the UK and indeed other english speaking countries they will agree with me.
A friend of mine - this indicates 1 person
My friend - Indicates 1 person
One of my friends - indicates 1 person
'My friend' is much more commonly used in the UK and I know Americans use that as well; a friend of mine is really long winded I never use that, you would perhaps use this if being very formal but it would be a rarity.
Hope that helps a bit