Aspies don't notice they're hurting others.
Sociopaths don't care.
It is actually worse than that.
In many cases sociopaths actually enjoy hurting others. They enjoy the feeling of power that it gives them.
At the same time, nobody feels more wronged than a sociopath when someone returns the favour and hurts them.
Well, I was looking at it from more of an Augustine perspective, where we don't do things unless there's some benefit to it for us. Us aspies are more like NTs here, where we would rather not hurt others, and try to avoid doing so. While the sociopaths you're talking about are getting personal joy out of it--they 'care' in the sense that it's an enjoyable thing for them, but that's for their own sake, not the sake of others.
Your last sentence is kind of where I learned a lot of my empathy from, which I suppose is another way this separates us from sociopaths. Once I've figured out why someone else is hurting, I know why I should behave differently. While sociopaths are more of the 'it's funny when it happens to other people' variety.