I don't hunt with either, but I use both for target shooting.
The '06 allows for heavier bullets. I've pretty much went to the '06 for 1000 yard shooting in windy conditions, but thats a bolt gun, in a gas gun where you're limited to bullet weight I'll stick to the 308.
The 06 wont die out, CMP GSM competition shooting is growing by leaps and bounds, when you're talking surplus rifles (1917 Enfield, Springfield, and Garand) you're restricted to the '06.
But for hunting, I don't think you'll see a difference. Some say you have the advantage of a short action with the 308 but I really don't think that's a real advantage, The Marines use a short action for their M40s in 308 where the Army uses the long action on their M24s. There is not accuracy advantage of the M40 over the M24. I built a dern accurate 308 on a Remington Long Action, I don't think I could build the same rifle on a short action that could be any better.
If I was to buy either as a hunting rifle, I'd look at the rifle itself and wouldn't let it being 308 or '06 determine whether I buy it or not.
Meaning lets say, I was to buy another Model 70, in 308 or '06. If they were the same price, and same rifle, I'd probably pick the one I could reach first.