Empathy is heartbreaking — you experience other people's pain and joy. Sympathy is easier because you just have to feel sorry for someone. Send a sympathy card if someone's cat died; feel empathy if your cat died, too.

Empathy was first used to describe how a viewer's appreciation of art depends on her ability to project her personality onto the art. These days it applies to anything you can basically "project your personality" on. When you feel what someone else feels, that's empathy. It's a good skill for doctors, actors, and characters from Star Trek:

Nearly all medical schools teach the importance of listening to patients and showing empathy. (New York Times)

"I've always thought of acting as more of an exercise in empathy." (Edward Norton)

In Star Trek: The Next Generation, Commander Deanna Troi was an empath: she could psychically sense other people's emotions. She experienced their emotions as they did.

Sympathy is an older word, from the Greek sympatheia, for "having a fellow feeling." It's a snuggly, comforting word. It's nice to get sympathy if you're feeling under the weather. To feel sympathy for someone is to feel bad for them:

This has already proved effective at drawing attention and sympathy. (Slate)

Police show no sympathy for "polite bandit." (Chicago Tribune)

So many dramas resort to cadging sympathy for their troubled characters by killing off loved ones. (Time)

If you're feeling empathy, you're in (em) the feeling. If it's sympathy, you're feeling sorry for someone.