"I don't care" means "it's not important." This can sound very rude though, depending on the context and who you say it to. It's a very casual phrase and I would never use it in a professional setting. "to matter" means to have value, so saying "I don't matter," would mean "I don't have value." This would be a very unfortunate thing to say and people would probably worry about you if you did.

I don't care- This indicates complete detachment and disinterest. It doesn't matter- To dismiss something,or simply express that whichever option chosen is irrelevant.

’’Care'' is an action. (In a ''I love you'' kind of situation) I care about you/I care for you. (For example, in a situation in a hospital/when someone's sick) I will care for you/I'll take care of you. There is an act of caring. ''Matter'' means ''significant''/''important'' more so than it means to ''care''. When someone says ''It doesn't matter'' It's like ''It's not important.'' There's a level of caring when you label something significant or important but that's not the meaning of the word.

Saying "I don't care" mean you don't have an interest in something. It can be about anything. I usually sounds very rude though so be careful where you say it. To say "I don't matter" mean you don't value yourself. People will become worried if you say this because they will think you don't value or love yourself.