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 wrote "I fear myself", which was marked as wrong (April 2020). I reported it ('My answer should have been marked as correct') so it should be added in due time. Many thanks to those who work hard to fix all the errors and add in all the alternative translations. :)
"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"...