Nothing magical about the G2 - it has a 3000mAh battery which is significantly larger than its direct competitors (S4, One) as well as having the improvements with the S800. The Note 3 actually has a 3200mAh battery while sporting a much larger screen.
The G2's battery is about 15% larger than the S4, but it can literally almost double the S4's battery life. The S4 just burns through mAh faster when doing the same tasks. The S800 counts for some, but when you look at the LG Optimus G Pro, which uses an S600, you still see more impressive battery life than an S4, and much slower mAh drain.
I"m convinced it's a combination of Samsung's AMOLED screens, particularly with applications that aren't optimized for them (meaning anything with a white background, which is a lot) plus the fact that Samsung's got 3 different procs in the GS4 and probably isn't properly optimizing for all of them. Everything I've read seems to indicate the Exynos powered GS4 can last a day with moderate use, so I'm thinking Samsung's Android build is optimized for that CPU and not Snapdragons. I'm tempted to obtain one for myself to see for myself, but I'm annoyingly happy with my G2 anyway.
You know, there's a zillion GS4 variants out there. It'd be interesting to compare an Exynos with one of the carrier S600 models, and compare both of those with the Google Play edition, and try to get a sense of hardware and software effects on battery life. Throw in that S800 model they sell overseas too.