For some design reason, the .308 cartridge is more accurate than the 30.06, so I have read. This is why after WWII people switched to .308 for matches, since .308 consistently beat 30.06.
Both have a diameter of .308. They both come in similar or the same weight bullets. The muzzle velocity of 30.06 at standard loads is just a little better, maybe 5%, but you can hot load .308's if you want more.
I think 30.06 is a little cheaper for standard hunting ammo. And all through this ammo panic, I have been able to find 30.06, but not .308 since people apparently are picking up .308 to feed their semi-auto black rifles.
The one thing I do like better about .308 is that there are several heavy barrel rifles available out of the box, while if you want a 30.06 heavy barrel, you have to go to the hassle and expense of getting a custom job. Heavy barrels mean better accuracy as the barrel does not heat up as quickly as a standard barrel, plus it is more rigid, hence less vibration.
If you step up to something like 300 Magnum then you are getting higher muzzle and terminal velocity, hence better range, but you are talking about a very significant increase in ammo cost. My neighbor took a deer at 500 yds. with one, and that was with a cheap scope.
I have two Remington 700's in 30.06, one low end and one high end, and I find them more than adequate; hell, I think 30-30 is great also, although it wouldn't do well for 300-400 yd. shots in the terrain you mentioned like 30.06 and .308 would.
Right now I have my eye on the new savage 10FCP-k in .308 which has a heavy barrel that is fluted, (comes with a bipod, accu-trigger, detachable 4 shot magazines and a precision stock) for about $800. I am not sure that it will shoot much better than my Remington 700's if at all, but I guess I have the impression that the black stock, fluted barrel, and bipod make it look more dangerous if ever I need it for TEOTWAWKI defense.
What ever you choose, I bet you are very pleased.