Both phrases mean that something doesn't matter very much to you, or that something isn't important, or doesn't worry you very much. There is a difference, however.
'I don't mind' is a fairly polite response, suggesting 'That's fine/ There's no problem/ I'm ok with that
'I don't care' can be seen as quite a rude response, suggesting that you are really not interested in something.
So, here's a question for you: If you are at someone's house, and they offer you a choice of apple juice or orange juice, which response would you give?