Here's an example. Your wife is trying to decide whether to wear a blue dress or a green dress for the party that you're going to tonight. She asks your opinion. If you say "I don't mind" (não me importa), this means that either dress is fine. "I'd be happy if you wear your green dress and I'd be equally happy if you wear your blue one". The subtext here is "I'm an easy-going guy and I think you look good whatever you wear". If you say "I don't care" ( não me interessa) it means "I have no interest in your appearance". This is a bad answer!

Use 'I don't care', when you have no interest in the outcome of a given situation/problem. Use 'I don't mind' when you want to signal your interest in the topic, but you are indifferent regarding the outcome. I hope this helps - I can imagine this being a bit confusing..

They are pretty much the same, I don't care is a bit more rude though.

someone can care any something in anywhere