Sympathy is feeling what they feel. Compassion is skillful action taken to alleviate suffering. Feeling bad for her is sympathy. An emergency medical tech administering saline is compassion.
Maybe I shouldn't answer this as a non native speaker myself (and contradict a native speaker on the way, sorry @crossfire) but from the etymology compassion (= with + suffering, German Mitleid) is feeling the pain of the other. Sympathy (similar + feeling, German, maybe Mitgefühl?) is not specific to suffering. You can feel sympathy for someone who's angry, jealous, enjoying themselves, etc.
Maybe I shouldn't answer this as a non native speaker myself (and contradict a native speaker on the way, sorry @crossfire) but from the etymology compassion (= with + suffering, German Mitleid) is feeling the pain of the other. Sympathy (similar + feeling, German, maybe Mitgefühl?) is not specific to suffering. You can feel sympathy for someone who's angry, jealous, enjoying themselves, etc.
It's ok, my definitions are more informed by my Buddhism.
Sympathy is neutral--feeling together
Pity is distancing oneself from the source of the sympathetic feeling
Compassion is actively engaging with the sympathetic feeling and working to heal the problem,
Sympathy is caring about someone's tragic circumstances. Compassion is a feeling of love for others that is charitable and willing to help no matter what. There's a sense of equality of heart no matter what background someone comes from. It's not about skill. It's about willingness.
Pity is distancing oneself from the source of the sympathetic feeling
Compassion is actively engaging with the sympathetic feeling and working to heal the problem,