The way I've had sympathy and empathy explained to me is that sympathy is when you recognize someone's situation is bad and acknowledge it somehow. Like, "Ah, that sucks."
Empathy is when you "feel their pain" more than just saying so. It hits you in the gut, I'm told. Kind of like when watching a sad movie and you break out crying, rather than just saying, "Oh, that was sad." It's sympathy + feeling the emotion (not just mirroring it).
It makes sense, I think. I've found I can "choose" to put myself in someone's shoes like that, but I still don't find myself automatically doing so like some people seem to do. I'm not sure if it's mirroring or actual "empathy" either. It also doesn't seem as "overpowering" as some people make it out to be. Sometimes I wonder if those people aren't full of shit, though... lol