"…compassion takes empathy a step further".
Yes, an important difference. As I’ve written about recently in HBR, three kinds of empathy are important to emotional intelligence: cognitive empathy – the ability to understand another person’s point of view; emotional empathy – the ability to feel what someone else feels; and empathic concern – the ability to sense what another person needs from you. Cultivating all three kinds of empathy, which originate in different parts of the brain, is important for building social relationships.
But compassion takes empathy a step further. When you feel compassion, you feel distress when you witness someone else in distress — and because of that you want to help that person.