Electrolytic capacitors use the chemistry to make a very high dielectric allowing the plates to get very close and increase the capacitance and decrease the size.

A cell in a battery is a capacitor then converts the charge on the plates into chemical energy and vice versa allowing much more energy storage and a flat operating range as the plates charge is replenished by the chemical reaction.

This article doesn't go into details but it sounds like the breakthrough is a much better dialectic then storing energy in a chemical reaction.