I think the reason Chassidish women don’t drive is the idea that a woman is supposed to be in the home (kol kevudah), and not galivanting around. They consider it to be loose behavior. I believe this is true after marriage, as well.
But don’t get me started on the topic of women drivers. Whenever I see one, I know my trip will now take 3 times as long. There is something about how they dart into a street in front of you like a cowboy (although they seem to be looking carefully, they invariably miss your presence, and that you have right of way), but as soon as they do this daredevil act of getting onto your street almost hitting you in the process, they begin driving 15 MPH below the speed limit, and are totally unaware that they are impeding traffic. Oy vey, don’t get me started.