Yes, there is not too much difference for me either! Except, in the situation when I am talking to someone about something one of my other friends did or said, I most often will use "a friend of mine...." But if I am talking about doing something with one of my friends, I will use "my friend" mkore often. For example, say, "I'm going to go shopping with my friend," or something like that. But, that is only me- I'm sure it varies by person! The meaning is not different. The only place I can think it is for sure one or the other is the phases: "My friend and I...." (never, "a friend of mine and I") "A friend of mine once said" (kind of a set phrase)

"My friend" is one single person. "A friend of mine" is one person taken from your group of friends. You're making it known that you have other friends as well as this friend.

for me, there's no diference, except "My friend" is more common than " friend of mine", which just sounds kind of classy/decent. I guess it depends on the worth mentioning. =P

Be the best Friend! :-)