The C-Stick is a key feature in A LOT of character's gameplans but it is especially important for Ryu. No one has outright said this yet but setting the C-Stick to "Attack" allows you to Up-Tilt lock opponents but slightly moving forward with the left stick and changing the times you press the C-Stick upwards. Furthermore, as Hanku Hirru said, if you use the C-Stick for standard attacks, you can input the true kill confirms much more consistently.
For example, if you have the C-Stick set to Smash Attacks (as is the standard), it is very difficult for you to input one of Ryu's kill confirms (U-Tilt to TSRK) because you must have the left stick in an upward position and then quickly input the Z-motion to TSRK. Instead, you can simply flick the C-Stick upwards and input the Z-motion just frames after. This will buffer the attack an net you HEAPS of kills.
The C-Stick's usage is mostly limited to Up-Tilt and F-Tilt (because of its disjoint) but these are very important to Ryu's gameplay. In regards to aerials, I would practice using the left stick to input a F-Air and then fast fall. Having quick movement through your left hand is very important, especially because it can be quite annoying to input a jump with the X and Y buttons and then transfer to the C-Stick. I've currently set the L-Button to Jump in order to do rising FADC and rising Tatsu which gives me extra recovery mixups.
