The phrase "of me" is simply a prepositional phrase. It is used the way a normal prepositional phrase is used.
This is a photograph/picture/shot of me.
Part of me wanted to run away.
The chubby man always said "There is more of me to love."
"Mine" is a possessive pronoun. The phrase "of mine" is used to indicate possession or association. This phrase can usually be replaced by "my".
John is a friend of mine. John is my friend.
This car of mine is a piece of junk. My car is a piece of junk.
Student or Learner