The .30-06 can drive a given bullet a bit faster and farther than a .308. The main reason the military adopted it was that the shorter length of the .308 is better suited to autoloading actions. The differences between the two cartridges in performance are more measurable than they are noticeable. Most of us would not notice a dime's worth of difference in practical usage.

If you like to plink and target shoot there's a lot more inexpensive surplus .308 ammo out there than there is .30-06 ammo.

For hunting, there is a huge range of factory loads for either that are readily available, and the biggest advantage you are likely to notice is that the .30-06 is better than the .308 with heavier bullets.