I’d say for this country the difference really is immaterial, but if your targeting large plains game, moose, bears etc then most of your good bullets start at 165grn, and if you use a tough controlled expansion bullet you want an impact velocity of over 2000fps ideally, and then your muzzle velocity becomes very important. In my .30-06 with a 175grn copper bullet I get that out to 500 yards, more than I’ll ever need, in a .308 you’ll get it out to 300. You may never shoot this far but a bullet going faster does stack the deck in your favour.
The .308 is great, it’s a much better all rounder for target and match shooting and some hunting, but I think the .30-06 will always be a superior hunting cartridge.