Our most recent stay in a Port Orleans resort was at Riverside, but up until then, we had always opted for French Quarter. Riverside got the nod this time around not necessarily because we wanted something different, but because (as noted by @
RandomPrincess above) the room only discount for that year did not include both resorts, only Riverside.
I can't say I was displeased with Riverside. The food court is certainly superior, and while I don't mind walking from French Quarter to Riverside to get to the in-hotel restaurant for a meal (the stroll along the river really is quite nice), it was nice to be able to leave our room and have it right there. I never really use the pools, but as a former kid, I can say that I think the theme of the Riverside pool looks a lot more fun. The grounds are definitely better, too. More land means more room for landscaping, and they use it to great effect.
All of that being said, I still prefer French Quarter. It has the better bar, I like the theme of the rooms a little bit more, but the two big draws for me are the intimacy of the resort and the bus service. The intimacy is kind of self-explanatory. It's a small resort, and none of the buildings are too far from anything. On the bus service: I'm not saying the bus service at Riverside is poor. It's better than the other large moderates resorts. because of the way the bus stops are laid out. You're never really too far away from your stop, or even your neighboring stop, for that matter. The thing that makes the French Quarter service better is that, since it's a small resort, there is only one set of stops that cater the entire resort, and sometimes, not always, but sometimes, those buses are exclusive to French Quarter. When they share with Riverside, there's no difference between the service at the two resorts (except for the fact that French Quarter is picked up first and dropped off first, giving the slight edge to French Quarter), but when French Quarter gets its own bus, the service is vastly superior, probably the best on property, since no other resort has that kind of bus to passenger/stop ratio.
That shouldn't be your only deciding factor, of course. If the ammenities are more important to you than the seclusion, Riverside is the superior choice. If you want a more subdued and efficient experience, and you're willing to walk a bit to get to better dining and pool options, French Quarter would be the better fit.