The confusion probably spouts from the fact that the capacitor stores equal and opposite charge on either plate, but the potential difference is what causes a movement of charge from one plate to another.
yes, since the capacitor stores equal and opposite charge, its net charge is zero... which is why I'm wondering what the capacitor stores? I'm guessing it's storing potential
There is an electric field between the plates. The capacitor stores energy in the electric field.
In the same way, a current through an inductor produces a magnetic field. The inductor stores energy in this field.