The .308 has a smaller case with less powder capacity and so generates a little less velocity with the same bullet. For this reason, it kicks a little less. The .308 does have a reputation for accuracy, but in reality youll find accuracy is more a result of rifle build quality than the choice between these two cartridges. There are plenty of sub-MOA .30-06 rifles out there, and the Remington 700 is a good bet for out-of-the-box accuracy. The difference in action length is insignificant for hunting purposes, in my opinion.
The .30-06 is loaded in more factory combinations of bullet type, weight and velocity than any other round. By choosing specialty rounds for the .30-06 offered from a host of manufacturers, you can duplicate the performance of magnum cartridges such as the 7mm Rem Mag or the .300 Win Mag. This is a big plus on large game. Also, as was stated earlier, if your gun makes it to your destination on an airplane but your ammo doesnt, you can typically find a decent selection of .30-06 hunting ammo at just about any hardware store in North America.
Go to federalcartridge.com and hornady.com to compare ballistics of the two cartridges, as well as offerings of specialty loads for both with enhanced performance. As a rule of thumb, recommended terminal energy for one-shot kills on deer-size game is about 1200 ft. lbs., and for elk-size game 2000 ft. lbs.
Bottom line: The .30-06 generates more velocity, hits harder, and has more factory hunting cartridge options. The .308s primary advantage is that it kicks a little less, if youre sensitive to recoil.
BTW, I wouldn't hunt grizzly or Kodiak bear with anything less than a .338 Win Mag (preferably a .375 H&H).