Some times these two words are confused in meaning. Yet they are very different.
Empathy strengthens connection but sympathy can move us to disconnection.
Empathy means that we can see the perspective of another person. It recognizes that the other person sees their perspective as truth. An empathetic person doesn't just recognize the emotions of another but acknowledges and feels with them.
Empathy recognizes that someone is in a dark place and does not just tell someone that "life will improve". The empathetic are willing to visit that dark place with the someone.
Empathy assures the someone that they are not alone.
Sympathy, on the other hand, will acknowledge a difficult situation but remains removed from the someone. Sympathy may encourage the some one to reach out but fails to realize their true state.
Sympathy does not connect with one in pain because we do not wish to feel that pain. Rather sympathy minimises the problem, making it less than what it is.
If we examine our conversation, many of us do this a lot. We do not want to become enmeshed in another's pain.
Empathy is prepared to acknowledge that perhaps we don't have an answer but that we are willing to listen and be a part of this journey.
Connection, a part of empathy aids in healing. Response, which is the prime ingredient of sympathy, does little.