Why use BG-E8 batteries over AA batteries?

Yeah, you are missing something. Two LP-E8s will last much, much, MUCH, longer than 6 standard AAs. For example, I went on a two week trip in New Mexico and the batteries lasted through that and a couple days after. To put it simply, LP-E8s can hold more charge than AAs. In the literature that comes with the BG-E8 (Batter grip) Canon states that using AAs will give you a shorter battery life. Also, LPs are rechargeable, as standard AAs are not, and by the time you bought 6 rechargeable, lithium ion AAs, you would be spending about the cost of a 3rd party LP-E8.

TL;DR buy the actual battery, keep six AAs and the magazine in your camera bag for emergencies.

but specifically I don't see why that is, the aa batteries have a much larger capacity, and also you can get 6 aa batteries rechargable for a third of the cost of a third party lp-e8.

I have always heard that AA's are best for an "emergency" situation, and that they provide inferior battery life when compared to original camera batteries (lp-e6, lp-e8, etc.). Also, a quick amazon search showed that it is around $20-$30 for 6 batteries, sometimes a little more (depends on if it includes a charger). They are nimh. These perform terribly in the cold weather, they self discharge, they take 3+ hours to charge (this is from the amazon page, most independent sites say 6 for AA, but it depends on the charger). Also, you can only charge 4 at a time, but you need 6 for you camera. You could buy 12 batteries and 3 chargers if you still really want to go this way, but that gets expensive and bulky. Long story short, us the lp's, and then throw some AA's in a pouch in your bag for that one day you forget to fully charge your camera

why buy something multiple times when you can buy it once?