Difference between: Don’t care / I don’t mind / It doesn’t matter / Never mind
I don’t care – this usage is informal and expresses indifference. It’s pretty innocuous, not at all impolite.
I don’t care – this usage expresses frustration or anger and often is an attempt to silence protest.
Indifferent: You are going to dinner with a friend who asks if you’d like Japanese or Italian. “I don’t care”, you respond, indicating you have no preference.
Annoyed, attempting to silence protest: Let me borrow LRV’s example above: ”I don’t care if you think it’s too early for bed, you are going now.” The child’s opinion is not important, dismissed, the parent is the boss.
I don’t mind. Means it doesn’t bother me, I give you permission. Can be indifferent, as in “Is it OK if I smoke?” or encouraging, as in “Is it OK if I kiss you?” Answer in both cases: “I don’t mind”.
Never mind often means “I was just talking about something but you didn’t get it so I am not going to repeat it. Let’s move on.” It could be innocuous: perhaps you were thinking out loud or mumbling and the other person didn’t hear the whole story and asks, “What did you say?” “Nothing. Never mind.” Another example: “I told you three times already but you’re not listening. Never mind!”.