Maybe you could think of 저는 - I 제 - my 자신 - self
translated literally it could be
I my self afraid
In this case it doesn't make a difference in terms of meaning due to the use of 자신, but with 무서워하다, the thing marked with -을/를 is the thing the cause of the fear, not the experiencer. So it means "to fear, to be afraid of", not "to scare, to make afraid". For example: 제 동생은 개를 무서워해요. "My little brother is afraid of dogs." (Not the other way around.)
무섭다 - to be scary 무서워하다 - to fear
I was wondering the same thing. Semantically, they appear to be the same, but... Perhaps there's a distinction I'm not seeing, or perhaps there's a similar distinction in Korean?
I take it to mean "self". Thus you get the sentence "I am afraid of myself"
"I fear myself" should theoretically be accepted, but yes, there is a distinction between that and "I scare myself". Obviously it makes little difference when the subject and the object are the same person, but consider the difference between, for example, "I scare the tiger" and "I fear the tiger"...
i heard koreans use 자신 more often in its meaning as "confidence" not so often as "self". the sentence to me made more sense in that regard as well...
so i gave "i am afraid of my confidence" a try, but...