I have had success getting a refund when moved to a 'less desirable' flight but, in my case, the difference was black and white. It was on a KHI-DXB ticket with Emirates - the cheapest fare specifically excludes travel on one of the 4 daily flights - EK603. The next fare up is identical except that it allows travel on EK603 and is about US$70 more. I paid this extra amount and was moved to the next morning's flight (EK605). I was able to get the fare difference back.
Unfortunately, it seems that weather delays seem to basically remove all liability from the airline for anything, consequently screwing over the passenger. If AA moves me to a cheaper flight, even if it was due to weather, IMHO I should be able to get the fare difference back if I gave AA extra $$ to be on that specific flight.
However, in all practicality, I don't think that there is any feasible way to implement such a system. If the passenger could prove that he paid extra and was moved, I think he might have a case in court. But then again, I'm not a lawyer...
Cheers,