I'm not actually AS, but I do have issues with empathy, so I'll throw in my $.02, if you don't mind.
Like ToughDiamond, I'm just really slow on the uptake when it comes to empathy. I have what I feel to be appropriate compassion and sympathy, but it may take me a while to catch on to emotional cues. I tell my students on the first day: "You've all had that instructor who could look across a room and tell who was getting things and who was struggling and needed help. I'm not that instructor. I want to help you, I would love to give you what you need, but I need you to tell me." This has worked out so well, I wish I could start every relationship off the same way. Things can get blown way out of proportion because the assumption is that I'm ignoring others' needs rather than that I just don't notice that they're upset. You've really got to hit me with a brick before I clue in to emotional context, and sometimes I still get it wrong. It would be so much easier if people would just come out and say, "Hey, I'm frustrated with you about something, can we talk about it?" or "Hey, it's really great to have you come to group things;" instead of just expecting me to understand how they feel.
This last one actually did get said at the last thing I attended by several people -- once they were drunk. It seems to be the only time inhibitions are lowered enough to actually give voice to things, normally.