Sympathy:- an affinity, association, or relationship between persons or things wherein whatever affects one similarly affects the other.
Empathy:- the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings, thoughts, and experience of another of either the past or present without having the feelings, thoughts, and experience fully communicated in an objectively explicit manner.
The terms are used more loosely than they used to be. In their pure form, both of these definitions imply a real suffering on the part of one person when another is suffering - crying, unable to move on, feeling depressed - vicariously experiencing their emotions. I would say that empathy is the more extreme. Someone with true empathy for another will sit with them and weep.
I would hesitate to claim that I have empathy for another, it would mean that I spent my days worrying about them, unable to think of anything else. It is enough to realise that they must be experiencing a dreadful time and send a little warmth and strength to them. It is NOT necessary for me to make a public scene of how much I am suffering for them and with them.
Does that make me hard hearted and callous?