Remember, first of all, that the structures of these two phrases are different. The subject of 'doesn't matter' is the word 'it', while the subject of 'don't/doesn't care' is a person. Secondly, the meaning is slightly different. It doesn't matter = It isn't important I don't care. = It doesn't interest me at all/ It makes no difference to me at all. Thirdly, remember that 'I don't care' can be seen as quite rude as an answer. If somebody asks you if you want orange juice or apple juice, they would be surprised - and maybe insulted - if you said 'I don't care' or even 'It doesn't matter.' Or in other words, they are much less polite than the Spanish phrase 'No me importa'. In a situation like that, I better response would be 'I don't mind'.

They mean the same. Some people may prefer one over the other, but I use them interchangeably. -US English

If someone doesn't care about something, then that thing doesn't matter.