From the out set I am biased toward the 30-06 for nostalgic reasons. Nostalgia weighs in on some of the opinions and that's ok. So let's begin by saying that the 308 is a fine cartridge and can hit targets far off and most of that depends on the shooter no matter which one of the two cartridges you are using.
With my Kimber 8400 Montana chambered in 30-06, I hit a coyote at 426yds (ranged shot) in a freshly plowed cotton field using a 165gr Game King HPBT. You can hit long shots with a 30-06 just as easy if not easier considering the flatter trajectory, less time getting to the target due to the velocities I was getting with that load moving a 3002fps. Is it a so flat that the 30-06 has an advantage over the 308? N0!!!!!!
When it comes to push and shove I can out run and hit harder than a 308 if I am hand-loading or using the light mag ammo. Big difference? NO!!!!! With the powders of today, the great bullet selection a person would be hard pressed to find a cartridge that is more versatile and provides more possibilities for hunting a wider range of game than the 30-06. Is the 30-06 more versatile than the 308? YES!!!
Guys like Page, Swank, Whelen, Waters did very well using the 30-06 and some of these men carried it and used it on big game in North America and Africa. Let's consider Hosea Sarber as an example, a Alaskan guide and warden, whom Jack O'Connor once said had killed more grizzlies than (Jack O'Connor) had ever seen using a 172gr bullet.
Also, there was Wendell Swank, a man who carried a 30-06 around East Africa collecting game specimen simply because it was the rifle most familiar to him. Oh, Swank did real good, taking the smaller animals with 150gr Sierra. But what was amazing was Dr. George Petrides who took just about everything from kob, wart hogs, and hartebeest, to lion, buffalo and hippos at ranges up to nearly 200yds. Not to mention Andy Russell, Myles Brown, Frank Golata, Alf Madsen, Jay Williams, Bill Rae and Bill Ruger are other men who have taken the 30-06 to Africa.
If it worked in Africa and in North America on big dangerous game then I am sure it has earned a place that the 308 will never occupy. Now add to that the new improved bullets and new powders that allows us to drive bullets out of the 30-06 at 300H&H levels.
In closing it was Stewart Edward White who said, "The 30-06 was as effective on lions as the big .405 Winchester." The 30-06 is like a good horse, that won't surprise you when the terrain and conditions change. The 30-06 is a very dependable, hard hitting, accurate cartridge that does not surprise you but just works hard and gets the job done. I like both the 30-06 and the 308 and what they are used for they are the best of the best, the most practical of the practical, the most used of the used in my opinion.