Sympathy = You thinking about someone and assuming what they are feeling, then feeling bad for them.
Empathy = You don't have to think about it, you just know what they are feeling. You react to how they acting.
Sociopaths could feel sympathy in some cases as long as they have felt a certain pain.
E.G. If a sociopath has been abused in their life, they might see a young child that's been abused and think about what that child might be feeling then feeling bad for it.
If a sociopath just sees a person in pain, and it doesn't relate to them in any way, then I doubt the sociopath would really feel anything.
Another E.G. Jason Vorhees kills people at will, regardless of whether they are good or bad. If they are invading his territory, he'll kill them.
However, if he was to walk across a small, depressive looking child being abused and hurt, he might well think about the pain he received as a child, and assume that that child is feeling the same thing, and may probably feel bad and take pity on the child.
This is different to the empathy that most humans feel. Most humans wont be raped in their life time, but they don't need to think to feel empathy for a woman or child who looks in desperate pain and misery after being raped. They will automatically feel bad for the rape victim.