308 was created to provide the military identical ballistics to its Garand 30-06/M2 ball ammunition in a shorter case, shorter action. For that reason--and loaded correctly--it will kill anything on the North American Continent using the appropriate bullet at appropriate ranges. (It would be more than a little light on Kodiak Grizzly, however.) It is ballistically very efficient and inherently very accurate.
The 30-06 does everything the 308 does and a bit more, a bit farther, a bit heavier. With its bigger case it is inherently more flexible at the medium/high end, but a little less efficient/flexible at the medium/low end.
Eaxmple: with 155gr bullet
Cartridge Vel IMR4895 Press Ball Eff
308..... 2,880 46.8gr 56,400 31.3%
30-06.. 2,880 52.5gr 51,213 27.9%
As you can see, the 308 is topped out at 2,880fps/2,850ft-lbs energy
The`06 has room to grow to 2,970fps/3,030ft-lbs for the same pressure (and 54.3gr of powder)
Question then becomes: "What do you need?"
If I had but one gun, it would be the `06.
If I had two, one would be the 308.