이별하다 and 헤어지다 have an overlap in meaning, but 이별하다 sounds more literary and poetic while 헤어지다 is more casual and has a broader uses. In daily situations, we'd use 헤어지다 99 out of a 100 times. Think of 헤어지다 as "to break up" or "split up", and 이별하다 as "to part (with someone)". Also note that 헤어지다 also has the casual meaning of parting with someone until the next time (우리는 다음날 다시 만나기로 하고 헤어졌다), unlike 이별하다 which usually implies permanent or at least a long term separation (그때는 몰랐지만 그것이 이별이었다 = I didn't know it then but that was our farewell.).

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