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).
What is the case for labelling them "Illnesses"?
I have consistently, over the course of my life, heard people describe sociopathy and related mental illnesses as being caused by a lack of empathy. This, intuitively, seems wrong, since that seems like a massively important brain function, that really ought to have a major and extremely visible effect on your thinking. Now, obviously it does have a serious impact (amoral behavior, etc), but it seems rather unlikely to me that someone like this really shouldn't be able to mask themselves as normal. (I'm also not sure why lack of empathy would make you want to dissect squirrels, but that seems like a side issue).
The upshot is that I'm seriously confused about what these mental disorders are, and how they work. Do these individuals have the ability to empathize but not sympathize? I'm not sure how that would work, but I'm not at all an expert on cognitive science. Is the standard explanation for these disorders just wrong? Are these people genuinely figuring out what humans care about by looking?
(As a side note, if it's the last one, has anyone considered getting a sociopath to work on FAI? Bringing someone who can't be trusted into an enterprise is a risky move, but if there genuinely are people in the world who have spent their entire lives practicing working out human emotions without feeling them...)