To empathize in this case is to appreciate someone's emotion. Empathy in a more general sense is the ability to put yourself in someone else's shoes and understand what they are feeling.
In this usage, it means that we, the public, feel for the victims of crime; we are capable of understanding their pain and loss.
To sympathize, as it is used here, means not just to appreciate someone's emotion, but to share it. This meaning is often used in compounds such as a "Communist sympathizer": this is a term used to mean someone who, while not themselves a communist, thinks that they may have some good ideas.
In many cases, sympathize can also be used to mean much the same thing empathize means in this context. The words are close enough in meaning that people are only likely to use them together when they intend to draw this exact distinction: they understand and appreciate someone's pain, but they do not agree with their response to it.