Porous manganese dioxide powders are synthesized by chemical precipitation from aqueous KMnO4 solutions in the presence of butanol-1 under mechanical stirring (using a magnetic stirrer) and ultrasonic treatment. The resulting products correspond to δ-MnO2 in phase composition and exhibit a hierarchically organized supraatomic structure. A comprehensive analysis of the experimental data using scanning electron microscopy (SEM), low-temperature nitrogen adsorption, small-angle X-ray scattering (SAXS), cyclic voltammetry, and galvanostatic charge–discharge methods show that the method of processing the reaction mixture affects the morphology and mesostructure of the resulting δ-MnO2 powder and does not significantly affect the values of specific capacitance and specific resistance of the electrodes formed based on it. At the same time, we note that according to the modeling data or data obtained by the galvanostatic charge–discharge method, δ-MnO2, synthesized under ultrasonic treatment conditions, allows obtaining electrodes with specific capacitance values 5% (simulation) or 9% (galvanostatic method) higher and with specific resistance values 11% (simulation) or 58% (galvanostatic method) lower compared to those for electrodes based on δ-MnO2, synthesized under mechanical processing conditions.