I'd imagine that would be very expensive presently. Plus, if the sheet they showed attached to the bulb powered it for five minutes, it contained 0.017 usable amp-hours at 2 volts or so (assuming I'm correct that it was a 20 mAh bulb in red, which it looked like to my eye).
Right now, your average AA battery contains 10,000 Farads or so. 162*aH = F, so that was a 2 F capacitor. Impressive in the extreme, but you'd need an area 5,000 times larger to equal one AA battery.
Capacitors have the disadvantage of having unusable charge in them. Any voltage under the usable voltage is overhead and essentially a waste. In the light example, when the voltage drops under 2 the LED wouldn't light--it wouldn't be strong enough to make the electrons jump the charge gap and create light. Electronics can be used to boost the voltage, of course, but even so there's a usable limit.
Supercaps usually have lower charge voltages than smaller caps, too. That isn't mentioned here, so I don't know with this technology, but to get the highest F rating you need a very large surface area very close to the other plate. That can lead to a discharge (a lightning stroke, or static discharge) that drains the cap at least partially and usually does damage to the cap at the very least. That limits the amount of energy you can put into the cap.
And you REALLY don't want to see what happens when you accidentally bridge a capacitor with high capacity. They're capable of extremely fast discharge...or extremely concentrated energy output over very short periods of time. You thought a lithium battery fire was bad? Protection electronics will be required. And I don't want to see what a mechanical failure on a high-F capacitor at any time, but particularly when fully charged.
Charging of a cap can be an issue as it essentially looks like an open circuit when discharged and starting to charge. If I use them in an environment where that could be an issue and overdraw a battery or circuit, you have to moderate the inflow to avoid damaging your power source (or, in the home, tripping the breaker). That's easily solved by a charger that won't allow more than a certain number of amps at a time, but does slow the potential charge speed.