Do you sympathize or empathize?

If someone’s family dog has just died, Do you empathize or sympathize with them? I never knew there was even a difference until my professor explained the difference. This was an important lecture to understand. Not everyone can empathize but everyone can sympathize. When you can empathize with someone, your advice or words are felt. Not saying that sympathizing with someone is not effective. It is easier for a person to connect with someone who shares a similar experience. I now know I would empathize with the person because I have lost a dog and know how emotionally painful it is.