Buddhist compassion is typically thought of as a feeling. A warm sense of positive regard. Sometimes it's extended to be an appropriate response - appropriate in the sense of helping someone to awaken. Some traditions have metta practices that involve wishing ourselves and others well. Empathy however is something much different. It's sometimes misunderstood as "feeling what the other person is feeling." The problem is that this starts with an assumption of sameness - I can feel what you're feeling. Psychoanalytic empathy starts with an assumption of difference - prompting one to ask - what does the world look like to this other person. It's an attempt to see and feel things from a different vantage point than your own that's equally valid.