"Don't worry about me" means don't be concerned because I will be okay. I would say this to a friend if I was walking home alone at night to reassure her that I'm fine.
"Don't care about me" is something you say to push someone away. In the example of walking home at a dangerous time, it tells my friend that she shouldn't care about whether I get home safe or not. I could say this if I don't want someone to have feelings for me. It's typically heard in circumstances where someone has mental health problems or unhealthy relationships because again, it is driving someone who cares about you to leave you alone, which can reinforce your sense of isolation.