My boyfriend has autism (not Aspergers).
He does find it difficult to see my perspective. It took a while for him to 'get' that I would have a different perspective on things (and therefore different emotions) to him. He tries, but I still come away with the feeling that he "didn't get it".
I have AS and I know that I also have difficulties in this area. I don't always respond in the 'expected' NT way, because I'm not NT. But it's not because I don't care.
So, for example, someone will tell me something and it might take me 3 or 4 days to realise that they were upset about it or what the 'correct' response 'should' have been. It will suddently just hit me whilst I'm washing up or something. I have some learnt responses ("I'm so sorry to hear that" when someone has bad news, for example) but I'm not able to 'find' anything beyond the learned response.
From what I see, my boyfriend is the same. But I know he cares about me and would never upset me or anyone else intentionally. And it worried him that he might unintentionally upset people when he realised it was happening.
I am going to challenge the 'no empathy' thing. Not for your husband specifically, because he might not empathise, but as an autistic trait generally. It is not. I can empathise and sometimes it is physically painful. I know that my boyfriend does too. But the delayed processing for both of us means it isn't always available immediately or presents in a way that is helpful and it can also be emotionally distressing for the autistic person. It might actually be that distress he feels at your emotional distress that he is rejecting, rather than your emotions per se.
Just out of curiosity, have you ever sought support, either IRL or online (e.g. different together website) to understand the impact of his ASD on your relationship and ways to manage it? I think it can be harder for people who are diagnosed as adults because their/our coping mechanisms haven't been developed through life within the context of autism and the ones we have aren't always helpful.
ASD is essentially a communication disorder, not a 'nasty bastard' disorder. So it shouldn't come as a surprise that there are communication difficulties. They can be overcome though, but it does take both partners to understand the difficulties, understand the impact of them on a relationship and then work together to put strategies in place to overcome them.
If either partner isn't interested in doing that, then that is more of a problem that the autism.