These illnesses are definitely not well understood, but you may have characterized sociopathy reasonably when you described it as lack of sympathy without lack of empathy. How they can come apart is indeed not intuitively obvious, but lots of the ways the brain works are unintuitive. Sociopaths seem to be the mirror image of autistics, who seem to lack empathy but not sympathy (autistics have great difficulty understanding the feelings of others, but seem as motivated as anyone to avoid hurting others when they can figure out how).