"Empathy" is the quality of identifying with or relating to the emotions of another person, normally on quite a deep level. It has elements of compassion and understanding. It is a quality which we hope those who we are really close to, as well as nurses, doctors and other people in the caring professions, will be able to show us when we need help.
"Sympathy", in general speech, is what we expect from everyone we know when we are in a difficult situation which we have not caused e.g. we are ill, or someone close to us has died. It is a quality which is not as deep as empathy.