What is the difference between Sympathy and Empathy?

While the dictionary definition may vary, I've heard them best described as follows:

Sympathy - Feeling sorry for someone and taking ownership of their problem, thereby trying to solve a problem that does not belong to us.

Empathy - Understanding the feelings of someone and trusting others to solve their own problems.

If one does not trust others, what does that say about oneself?

What fears might be driving us to not trust others, so that we can empathize with them and truly BE WITH them...?