your true face takes face as metaphoric for character. It suggests that a person's character has been revealed to be other than what it appeared to be previously. The truth about the person has come out. A related expression is "to show your true colors".
A true story is true because it actually happened. It records facts. Otherwise, it is called a work of fiction. Or, if you are telling falsehoods, it is called lying. You can say, "I assure you I'm not lying. What I'm telling you is a true story."
I would use "true story" because a real story would be the opposite of an artificial story, and that doesn't make sense to me.