自分 is used for "myself, oneself, one's" etc and as a personal pronoun, like 私.
自身 is used for "oneself, one's", but cannot be used as a pronoun. It also functions as a "strengthening" word sometimes, eg. 彼自身 (himself).
But yeah, it's difficult to explain since they're quasi-synonyms, you could check how they're used and their differences at
http://eow.alc.co.jp/.