Both sympathy and empathy are about feeling concern, care, understanding, and wanting to support a person who is going through something unfortunate and/or unpleasant. The difference as I understand it is that a person who has never experienced that situation themselves can only feel sympathy while a person who has gone through it themselves will feel empathy.
For example, John's dog just passed away. Eric never had a pet before but he understands that losing one is tragic and so he feels sympathy for John. Mark had a dog pass away a year ago, and so he knows exactly what John is feeling. Mark feels empathy for John.
Saying, "I have sympathy for you", is saying that you care, but admit that you don't have the experience to feel what they're feeling. In some cases that might be the more respectful thing to say because the person suffering might believe in that moment that no one can truly understand. Empathy would seem like presumption.
