This answer has been converted into an official blog post, check it here: https://talk2.wisen.com.au/2016/12/08/p ... y-battery/
Hello lowpo,
Short answer
There are a few differences... probably the most important is the size versus capacity as you mentioned, but there's also the chemistry which will affect the cell voltage as well the discharging curve, which I'll try to cover at the "Long Answer".
- AAA 1200mAh
- AA 2700mAh
- CR2032 250mAh (3.0V)
Note that the CR2032 is a 3.0V battery, so 250mAh here can be compared with a 500mAh 1.5V Alkaline battery. The other point is that the coin cells, in general, are designed to delivery only a few milliamps of constant current. Constantly draining more than 0.5mA will significantly reduce the life of the battery. The alternative is to consume the energy via pulses, for example, sleep --> short run --> sleep...
I personally like to stick with a pair of AAA or AA (if the size permits), just because the discharge curve is very predictable and easier to know how much juice remain in the battery by monitoring the cell voltage. The Alkaline batteries are also cheaper and can be found anywhere.
Long answer
As I mentioned on the "Short Answer", there are more than just the battery's capacity and size. Depending on your project, carefully choosing the battery could improve how long your board can run. This can be very important, specially if your hardware will sit behind a dry-wall or in a difficult/remote access location.
To compare batteries, it's better to compare first the ones with the same chemistry. For this answer I'll compare only the non-rechargeable options as the rechargeable batteries tend to have a very high self-discharge rate, unsuitable for long running projects (1+ years) - more to it can be found here:
http://batteryuniversity.com/learn/arti ... _discharge.
Alkaline
Most of AA, AAA, C and D batteries sold these days are Alkaline. You still can find some brands labelled as "Heavy-duty", and although the name sounds like a good thing, they normally have smaller capacity than Alkaline ones.
The Alkaline cells are sold as 1.5V but they actually start their life at around 1.55V to 1.6V and are declared dead when the voltage drops down to 0.8V. Saying that. only few products, like the Whisper Node, will drain the battery down to 0.8V, most will stop working at 1.2V, discarding good part of the capacity that still on the cell. Just measure the old batteries from the TV Remote or a kids toy when replacing it.
Here the Energizer Alkaline AA battery datasheet link:
http://data.energizer.com/PDFs/E91.pdf, you'll be able to see that the battery is able to delivery almost 3000mAh when discharging at 25mA down to 0.8V.
The voltage drop on the Alkaline batteries is quite significant, which is bad, but also useful as it's easier to determine the current cell's capacity.
Non-rechargeable Lithium (1.5V)
Those days you can also find the non-rechargeable lithium batteries on standard AA and AAA sizes @ 1.5V, as a drop-in replacement for the Alkaline batteries. They normally have 5%-15% more capacity when compared with the Alkaline cousin, but the biggest advantage is that this kind of battery can delivery the total capacity at high discharge rate - which is great for a camera or high drainage toys, but doesn't offer much more energy for low-draining applications.
Here the Energizer AA Lithium battery datasheet:
http://data.energizer.com/PDFs/l91.pdf. An interesting fact is that those cells have up to 20 years of shelf-life, indicating a very low self-discharging rate. This battery also can operate at a wider temperature range: -40°C to 60°C.
This kind of battery will have an almost flat voltage along its life, which can be challenging to determine the cell's current capacity before it's already getting close to the end of the life.
Non-rechargeable Lithium (3.0V)
Those kind of batteries are commonly found in two packages: coin-cells and small cylindrical - the second normally used in cameras/flashes.
Both have stable voltage during their life and only start to have some significant voltage drop after 2/3 of their life. Keep this in mind when using the voltage as an indication of the cell's current capacity.
Being a Lithium battery, the working temperature range is pretty good: -40°C to 60°C, as well they have very low self-discharge.
Different from the AA Lithium batteries, those are designed to be used in "Pulses". For example, continually draining more than 0.5mA from a CR2032 will significantly reduce its life and the battery will not be able to delivery the rated 250mAh. The same applies for the CR123a or CR2, but at a different scale.
CR123a -
http://data.energizer.com/PDFs/123.pdf
CR2 -
http://data.energizer.com/PDFs/1cr2.pdf
CR2032 -
http://data.energizer.com/PDFs/cr2032.pdf
To efficiently drain this kind of battery, the application needs to have a very low constant drainage and might some high current pulses from time-to-time. This matches exactly what most sensor nodes will do: the product will sleep for a long period and turn-on/transmit data for only a few milliseconds per cycle, giving enough time for the battery to recover.
If you have space constraints, but at the same time need good energy density or low self-discharge, this kind of battery can be the answer.
Primary Lithium Thionyl Chloride (3.6V)
Those batteries are not available everywhere, those are specialized cells, but they are an excellent high density energy option. They have an incredible characteristic of keeping the voltage stable all over their life. It's very common for products designed to use this kind of battery don't have any kind of power regulator.
Common applications are remote sensors, beacons, sealed products, etc. If you're curios, have a look on the datasheet links below, just don't get too excited as a single AA cell costs no less than US$ 9.00.
Equivalent AA size:
http://www.saftbatteries.com/force_download/LS14500.pdf
Equivalent D size:
http://www.saftbatteries.com/force_download/LS33600.pdf
Li-Po
There are some details about using Li-Po with Whisper Node at this topic:
http://talk2forum.wisen.com.au/post20.html 
