It basically means very similar, I found it quite challening to explain the differnce between them, so I will give you some exmaples using those words.
그는 자기가 잘생긴줄안다 = He believs himself handsome
그는 자신이 잘생긴줄안다 = Same meaning
그녀는 자기가 잘못한걸 알았다 = she knew herself that she did wrong
그녀는 자신이 잘못한걸 알았다 = Same meaning
As you can see there is not much difference. What you need to be careful with is 자신"이" and 자기"가"
Was it helpful? I hope this helped you :)