It depends on what you're hunting, and how you're hunting. If you mainly plan to hunt deer at moderate distances, then the 308 would be the better choice, mostly because of the lower recoil. If you plan to hunt bigger animals like elk or moose, then you'll want larger bullets, and the 30-06 would do better for you. If you plan to do both deer and larger critters, then the 30-06 is the better choice, just for when you go after the bigger game.
The military didn't change from 30-06 to 308 because the 308 is a superior cartridge for everything under then sun. They changed for they're specific purposes. Less recoil, more ammo carried for less weight, and it kills humans [which are pretty small] just fine for the ranges encoutered in battle most of the time. Also there was something to do with the advent of fully automatic actions, and keeping ammo consistent throughout, I think. Just because the military changed to 308 does not mean that it is a better HUNTING cartridge. Two totally different scenarios and different crtieria in what you want your round to do.
The difference in the real hunting world between the two is approx. 100-150 fps in factory ammo with the same weight bullet. When you reload you can spread that difference to over 200 fps. That difference in velocity will extend the MPBR of the 30-06 by 30 to 50 yards beyond the MPBR of the 308. That means if I judge the deer to be at 300 yards and its really at 330 yards, with the 30-06 I'd still have a kill, but with the 308 I'd miss.
But if the deer is at 200 yards, then both will kill just fine.
The 308 has less recoil and can go in a short action. It can kill deer, elk, moose, or whatever. The 30-06 has more recoil. You can get it in very light guns, but then the recoil is even greater. Personally, I don't mind the recoil of a 30-06 in a light gun, but I'd certainly understand if someone didnt like it, and developed a flinch. The 30-06 will also kill deer, moose, elk or whatever, but can do the larger with more aplomb because it will shoot the heavier bullets faster, and with the lighter bullets it will extend the ranges as well.
My own opinion is that if you want a short action hunting round for deer, the 7-08 is the best choice. If you want a round dedicated for elk or moose, the magnums [30 cal or 338] is best. If you want an all around, do everything cartridge, the 30-06 is king. If you want a long distance target round 6.5-284 is the way to go.
