Sympathy or to sympathize with someone means to feel sorry for their misfortune, tragedy, pain, or loss. That sounds simple enough. But it gets a little complicated. Another definition of sympathy is to share an understanding or common feeling. Having a common opinion or perspective on a matter.
Empathy or to empathize is the ability to understand and share the feelings of someone because you can relate to their experience, or imagine you can. Empathy focuses on walking in someone else's shoes either because you actually have had a similar experience or a strong ability to imagine the magnitude of the experience.