Just me but I think people forget that empathizing and sympathizing are two different things. Figure empathizing means you know how they feel and can to a degree feel for them, sympathizing means you can feel for them and their situation. A lot of times I've found people want you to empathize with them quietly and in private and perhaps give them a shoulder to lean on when they need it. But keep that empathy to yourself or between the two of you. Sympathize can be shared in public for many times it's a common cause or complaint and can foster a sense of public support when asked for. Bad thing though with both is they can be destructive as well as it can be seen as belittling of the person or even making the person appear weak depending upon how you present it.
Figure we may empathize and sympathize with the disabled person but have them resent us at the same time for it makes them feel less or weak by our very actions. They fight to be seen as equal or strong in the public eye, and their own sense of self worth and any such display in or can be seen as counter productive by them.
Of course just my own take on the two words and how I see them so others may use them differently.
Yes. I understand that empathy and sympathy isn't the same thing.