Hi,
Unclejed, that's a very good point. I would vote that one of the best points made if we had a voting system for that here
As to a more direct comparison for the battery vs capacitor, we can begin with calculating the maximum energy that can be stored in an air capacitor and then compare that to alkaline cells. I took the liberty of doing most of this already so i'll show mostly the results here and the facts behind the calculation. Also note that this is still an approximation because there are various assumptions such as air pressure and temperature and moisture content, to name but a few.
Knowing that the capacitor will arc over if the voltage is too high versus the distance between plates, the maximum energy storage in air is about 17.7 Joules per cubic meter. That means that a capacitor that takes up 1 cubic meter with air as dielectric can only store at best 17.7 Joules.
Now if we look at one alkaline cell, it's pretty small, taking up a volume of about 1/120000 of a cubic meter. Yet it holds approximately 10000 Joules (or more) of energy when brand new.
So for the air capacitor, we have a ratio of at least 500 to 1, and that's even before we look at the entire volume!
Noting that we can fit about 120000 AA alkaline batteries into a space of 1 cubic meter, that means we can store:
1200 million Joules of energy in 1 cubic meter if we use alkaline cells instead of air.
Now to be fair, we have to realize that most capacitors used today have a dielectric. And that dielectric boosts the energy stored in a given volume because it is denser, and also helps to prevent the plates from arcing over. The very fact that it can store more energy per unit volume means a lot, but the added benefit means we can charge the capacitor to an even higher voltage within the same volume. So for a mica capacitor (dielectric constant around 5) we end up with a storage ability of about 2200 Joules per cubic meter.
So for alkaline we have 1,200,000,000 Joules per cubic meter,
for air we have about 17.7 Joules per cubic meter,
and for mica we have about 2200 Joules per cubic meter.
Note even with the mica we are still way behind the alkaline as the alkaline holds more than 500000 times more than mica, and about 68 million times more energy than air.