Allows for a protective case
More combined capacity than any extended battery short of the Zerolemon
Potentially as much battery as you want if you get the Seidio one with extra batteries
No protection, except with the Seidio one, which makes for a huge device
3rd option : External power pack, such as the Patriot Fuel+ units (My personal choice. I own one of the 6000mAh units) or Aluratek's even bigger unit.
Could also potentially be combined with either of the above.
Potentially more battery than any of the above options
Doesn't make your phone bigger
Allows for any case you like (e.g. Lifeproof or Armour)
Requires pockets to carry it
Clumsier than the battery case when charging
Requires a second charger for the pack if you want to charge both your phone and the pack
You need to remember to charge it occasionally.
Samsung Galaxy S3 LTE (i747)- Rooted, stock firmware, safely ensconced in my Lifeproof Fre
HTC Incredible S - S-off (rev hboot) - returned to stock ROM and handed down.
HTC Wildfire S - BL unlocked, unbranded, gathering dust.
Archos 32 - SDE flashed, rooted, gathering dust.