By using both, you can trajectory DI one way using the control stick, while overriding the ASDI you'd normally get with that direction on the control with a different ASDI you want instead by using the c-stick for it.
http://www.youtube.com/watch?v=Ei-iOIN32i4
You can trajectory DI a hit you can't avoid to get the lowest angle of knockback (and with it minimizing the amount of vertical knockback), while also ASDIing straight down into the stage so the ASDI will cause you to hit the floor and be able to tech until much higher damages than ASDIing down alone.
It works great for things like recovery move landlag and other very laggy moves to avoid getting KOed or escaping a move that would otherwise set you up for a combo. For example, a lot of the cute combos mentioned in this thread can be escaped with it