Hello Superorb,
Let's move from talking about single cells to talking about how they are used in a flashlight...
Many LED lights contain a circuit that regulates the voltage and current going to the LED. Depending on the requirements of the regulation circuit, a number of battery combination's are available.
If the circuit steps the voltage up, you can run an LED off of 1 Alkaline or NiMh cell. If the circuit drops the voltage, you can run an LED off of 2 Li-Ion cells. Some circuits do a combination of both.
LED's require a certain voltage to work for flashlight use, and they work best at a constant current. LED's will glow at low voltages, but we expect a little more than a glow from our flashlights.
If we look at a battery pack to power a flashlight that has no voltage regulation circuit in it, I think you will begin to get this concept.
We can take 3 NiMh cells, put them in series, and this battery pack will have a voltage of around 3.6 volts, and, in the case of the Duraloop cells, a capacity of 2000 mAh. If you set 3 NiMh cells end to end on a table, you will have an idea of the shape of this light. It will be long and narrow.
On the other hand, we can take 1 Li-Ion cell and end up with about the same voltage. There are many options on capacity ranging from the R-CR123 cells at around 600 mAh, to 18650 cells with up to around 2500 mAh. In the case of using an R-CR123 cell as the battery, the flashlight would be a little fatter than the other one, but much shorter. This ends up being a very compact package for a flashlight.
That should give you a starting idea about size differences, now let's look at available energy.
The battery pack made up of 3 NiMh Duraloop cells would have around 7.2 watt hours. The battery pack made up of 1 R-CR123 cell would have around 2.2 watt hours.
Now, if your light uses 3.6 watts, the light powered by the Duraloops will last around 2 hours, and the light powered by the R-CR123 will last around 36 minutes.
Now you have to look at how you are going to use the light. The light powered by the R-CR123 battery will easily fit into your pocket and may be readily available when you find yourself needing a light. The other light will most likely end up on a shelf or in a tool box and when you need a light you will have to go find it before being able to use it.
Many people find that they use both types of lights, and as you can see, there are an infinite combination of variations between these two examples.
Tom