They're close in meaning, but compassion is deeper. One way of looking at it is to think of 'sympathy' as understanding another person's situation, and 'compassion' as really feeling it. Compassion is a much stronger word, and so is used much less.

I think sympathy implies that you like that person. Compassion is that you feel for that person. You probably wouldn´t feel sympathy for someone who´s done something very wrong, but you can feel compassion if you understand the reasons or feelings that he went through to do that.

Compassion comes from the heart. Sympathy is decided in the mind. Empathy is both.

"Sympathy" represents an outward expression. "Compassion" is an internalized feeling. Someone might extend sympathy in words, but not feel it inside. People with compassion may take action and not stop at words.