I haven't noticed a difference regarding housework between NT vs ASD people I've lived with. Usually we prefer different chores and just divide up the remaining ones between us or have designated days for things (whatever works for our schedules). It's pretty regimented from the start, so everyone knows exactly what is expected and is in agreement (saves having arguments over supposed misunderstanding later on). If one of us is working loads of overtime and not getting home until stupid o'clock at night, then the other helps out a bit more and vice versa. If both are doing 80+ hour weeks, then we just try to cover the basics and don't expect everything to be perfectly tidy for a while. We just use common sense. Some people are incredibly lazy though, and I would avoid living with that sort of person.
I also can't say I've noticed a difference in standards between the men and women I've shared a house/flat with. If anything, the men tended to really enjoy cooking, compared to women who just eat to survive and will throw together a salad if we're on our own. So although I can cook, I usually ended up doing other tasks as it was more of a hobby than a chore for them. I love the smell of clean washing and the fabric cleaner/conditioner products, so I'm always happy to sort out the washing. I also have a weird enjoyment of making beds (again, because of the clean material smell) so those are usually my designated tasks.
A couple of women have been very messy (fortunately only temporary housemates) and covered the bathroom in products, hair, makeup or dye on the towels, half melted candles (I think?) and other junk that would have eventually annoyed me in the long run. Although my main issue with living with those women was when they constantly used up food or various other items (toothpaste, soap, etc) and didn't bother to replace it so I'd have to walk to the shops late at night so I'd have food to eat or be able to wash when I got up for work the next day! And they either ignored any complaints or acted like it wasn't a big deal that we were late for work. We quickly agreed to kick them out of the flat, so luckily it wasn't a problem for very long.
Selfish behaviour like that annoys me more, as those people just don't give a damn about anyone else and the effects of their actions and clearly think the entire world revolves around them. I have no issue with people borrowing my stuff, but when it completely disrupts your schedule because they don't bother to replace it or even tell you it's gone, then that drives me nuts. Far more than a messy house (which I can at least wait until I get home to deal with). If you have to be somewhere by a set time (i.e. the office) and their laziness make you late, that's far worse because you can't plan for it.