To be frightened of something means to be scared of something for a shorter period of time or to a smaller degree. Generally, when are frightened of something, you can still be in the presence of the thing or object without experiencing emotional harm.
Ex: Justin knew his mum was frightened of clowns and promised her they would not be at the fair. This little white lie got him in a lot of trouble when his mum walked into three clowns while waiting in line for the lavatory.
You can always be frightened by or scared by something. For instance, if you are frightened or scared of storms, you are afraid that they exist. However, when you are frightened or scared by a storm, it means you see a storm or you are in a storm and it is scaring you. However, the preposition by cannot be used in the same way with the word afraid.
Ex: George was frightened by the thunder and lightning that was starting outside. But he had to show bravery and drive in the storm if he was going to make it to the airport on time.
Ex: While I was a little frightened of the hairy creature, I was
more afraid of what Josh would do when he learned that I was the
one who put it in his bed.