For different supercapacitors, i.e., double-layer capacitors (EDLCs) and pesudocapacitors, the working principles are different. So you have to look into the working principles to explain the difference in capacitor performance from amorphous vs. crystalline materials.
Further to my earlier answer, for metallic oxide the higher crystallinity is always preferred. For organics mostly amorphous materials are used for fast movement of ions.. One important propery of organic crystalline material is the charging and discharging is fast due to ordered path for charge mobility. Another way we can look at is the ordered arrangement of crystalline materials is like nanorods which leave gaps between two rods or crystalline domain for ions to move fast and enhance the capacity.
I'd like to contribute to this discussion, can you specialists quantify your answers in Farads or alternatively as Ph.D. Samui did by capacitor type. electrolytic etc.
I am unfamiliar with supercapacitors, but have industrial experience with Capacitor wrap tape. The greatest commercial Farad rated capacitors I know are in audio amplifiers (tube type). I suspect there are specialized custom builds in either Astronomical equipment where the signals are small, or spaced based equipment where they are cold and the signal is clean from thermal noise.
In contrast to crystalline materials, amorphous materials have large number of structural defects (e.g. vacancies) which facilitates the diffusion of electrolyte ion and electrochemical reaction. Further, amorphous materials experience isotropic stress and strain of charging-discharging which helps in better cycling stability. However, low charge carrier mobility in amorphous materials is a considerable limitation.