I took a girl out recently, it was a little awkward at first but I couldn't put my finger on why. As she started to feel more comfortable with me she opened up and told me she has AS, which immediately explained a lot of the behavioural stuff (lack of eye contact, quite direct to the point of rudeness, missing some standard social cues). I immediately felt more at ease and we had a thoroughly enjoyable evening...
...right up until the end of the night when she invited me back to hers. I said that while I'd be happy to, I didn't want her to expect too much, as she had indicated a few times she wanted something serious while I have just come out of a 19 year relationship and am taking dating slowly - my concern was having sex that night would make her feel that I could be more to her than I was able for. Anyway, she didn't react well to that, presumably because she took it as me rejecting her offer. She left and I texted her after wishing her well and saying I hope she finds someone.
The following morning she messaged to apologise for overreacting and offered to have me over and make me dinner. We haven't done this yet but we've been messaging and it's all going fine, but I am still a bit concerned she wants more than I'm able to give.
In summary, dating a girl with AS isn't much different than dating a girl without AS.