Using Brians numbers for a 150 gr bullet the 308 gives up about 2.5% velocity but uses about 14% less powder and has about 12% less recoil. If you also factor in that with modern bullets there is simply no need for the heavier bullets that used to be a advantage in 30-06. A 150 gr, even 130 gr premium bullet from a 308 will kill anything on this continent.
I've hunted with a 30-06 since the 70's, have several that aren't for sale. But my 308 is 2 lbs lighter, shoots just as well and goes hunting a lot more often now. I cannot imagine any animal ever knowing the difference.