Self-centered means the majority of time, you think and focus on your own self-interests.
Selfishness on the other hand, is a focus so strong on one's own self interests that it takes precedence over the well-being and interests of others.
An example of being self-centered would be spending the majority of your time working for your own benefit and no one elses. It's not necessarily harmful, but it focuses on serving your own interests and no one else's.
An example of selfishness on the other hand, would be having a large surplus of money and food as a result of working for your own benefit, and not being willing to share any of it, even with those who are in desperate need.