The difference between "afraid" and "frightened"

Being "afraid" means that someone is feeling/experiencing the feelings of "fear". It's a mental or a psychological state. It could result from psychological or mental reasons. 

Being "frightened" means that something made someone afraid (i.e. someone is afraid of a specific "fearful" person/being/thing). More related to "sensuous" things: things that you can hear or see.

Compare the following:

I'm afraid of the exams. 

He is afraid of failure.= He fears failure. = He doesn't want to fail.

This huge dog frightened me. I couldn't enter the place.

He is frightened by the dog. = The dog makes him afraid. = The dog scares him. = He is scared of dogs.

Please note that:

Scared  is less formal than frightened.