What's the difference between saying "I don't mind" and "I don't care"?

"I don't mind" means: "I care about the issue or concern you raised. However, your proposed action is OK with me."

"I don't care" means: "I do not care about the issue or concern you raised."

Example: Suppose you have a TV show on. A friend comes up and asks if it's OK for him to change to another show.

"I don't mind" means: "Thank you for asking. While I might have been watching the show, it's OK if you want to watch something different."

"I don't care" means: "I wasn't paying any attention to whatever was on. And I won't pay any attention to whatever you turn it to. It's meaningless to me."

See the difference?

You can care about something being done but not mind if it is being done, but the reverse isn't true. Such as, you can care if someone tosses their loose change into the street, but not mind that they do so if a beggar is there to pick it up.

I don't mind -means= OK, no problem. This one is a positive response.

I don't care-means=OK and stop bothering me. This one is negative.

When you say "I don't mind" you're implying that whatever they want to do wouldn't bother you and you'd be fine with them doing it, but if you say "I don't care" you're saying that you have no interest at all and you don't really want to be bothered.