Pop. Pop EVERY time.
I have stayed at both Pop and POFQ. I love the Pop food court, although it is true that it does get busy, but the queues are the same anywhere you go. I waited in line longer at POFQ than at Pop many times!
Be aware of the difference between POFQ and POR - the foodcourt at POFQ is small and has less choice than POR or Pop.
Riverside has a make your own pasta bar that people really enjoy but this is not there at French Quarter. FQ does have beignets, but I was a bit disappointed by how greasy they were. Also be aware that FQ seems to have its own recipe for standard stuff - FQ pizzas are not the same ones you get elsewhere on property eg in Pop, pizza planet and every other counter service I can think of! The dough and sauce are different (and to my taste, not as good). Other things were subtly different, which may be to your taste or may not be. I was hugely disappointed in the food court at POFQ.
At FQ I missed having chocolate croissants to eat at breakfast that I can get at Pop, and found the whole baked goods very limited at POFQ. I also prefer the wider range of choices that you get at Pop.
Pop has better baked section (pastries, cakes, cookies), better chilled section (salads, fruits etc) and a wider selection of hot dishes (flatbreads, asian noodles, pastas etc).
I agree with the pp who said POFQ has the better rooms, Pop the better food court. To give you an idea of how I feel having been to both, this December we are going to Pop again
Hope this helps your decision!
Miranda x