If you look at their patent, it is a slightly modified Ultra-capacitor, running at 4000v (to keep leakage current down to a minimum)

And some semi-conductor modifications to the electrolyte to reduce the leakage current further.

They are still quite heavy.

The main benefits over a conventional battery are energy density, dump current, supposed life span and charge time (no different to existing capacitors really)

The benefit over a standard capacitor are its supposed capability to store charge for long periods of time. This is where many of the professional arguments are focussed as EEStor after approximately 2years since they announced this device, have to the best of my knowledge still not released any data or allowed a device to be tested.