Hi,
I agree with everyone else that this is down to personal preference. I have stayed in Riverside a few times and French quarter once and enjoyed them both.
Frech Quarter:-
The pool is great - I loved the waterslide and the theming and the poolside bar.
I really enjoyed Bonfamilles (now gone) and Scat Cat lounge (is that still there?)I preferred the food court in FQ to the one at Riverside.
The bus service seemed better we always got seats on the buses - when they got to Riverside they were quite often standing room only at busy times (or full) and we always got dropped off first on the way back.
You are always close to the food court even in the building furthest away - in Riverside you can be quite a way from the foodcourt/reception etc.
The whole resort feels more intimate and is very pleasant. The only down side is that our room was quite noisy -3rd floor - and it was the plumbing and A/C that made the noise.
Riverside
I love the look of the elegant buildings and strolling around the resort or hiring a surrey bike.
The buildings are only 2 storeys - not as many stairs at the end of a long day.
I couldn't stand the bar 'entertainment' - this is a personal thing, but they had a bloke in singing and playing the piano most nights - it was very poular and very loud - this carries down into the foodcourt and Boatwrights - most people loved it, but for me it was too Chas n Dave and there was no escaping it!
You can be a long way from reception, main pool and foodcourts - this has advantages and disadvantages, but with kids I would say chose your room location carefully.
Both resorts are wonderful and I definitely prefer them to CBR - catching the boat to Downtown Disney on my first night there has become a ritual - and best of all its free.
The only other thing I would add is IMHO it isn't worth upgrading to Riverview room in either resort - you don't have a balcony and you can walk around the buildings to look at the view.
You will have a great time wherever you stay.