Qual è la differenza tra I don't mind e I don't care ?Sentiti libero di rispondere con un esempio.

These are very similar but "I don't mind" basically means you do not care about something or something doesn't bother you. Here are example sentences :
I don't mind driving on long road trips.
I don't mind dogs as long as they are friendly.
Or an example conversation would be:
Person 1: Sorry my room is so messy!
Person 2: Its ok, I don't mind.

I don't care you can use similar to "I don't mind" but you should say "about" or "if" afterwards. But it has a slightly different meaning, its more harsh and it means it doesn't bother you if something happens or you don't even think about it. Examples:
I don't care if you want to stay up all night!
I don't care about anything right now - I'm so sleepy!
Person 1: What restaurant do you want to go to?
Person 2: I don't care!