I think the shadow that the 30.06 casts over the .308 is mighty small in a practical sense, particularly in the 180gr and under weights - which I would imagine is 98% of the usage in those two cartridges. For midwestern whitetails I mostly use a .308 and out west for elk, I use an '06 - mainly because my .308's (excepting a Ruger 77 VT) are short barreled (handier in the treestands and thickets) and my '06's are sporters with longer barrels more suited to the west. Also I use lower powered scopes in the midwestern woods, more magnification out west. Other than because of what rifles I am using, I don't believe the .308 gives up much to the '06 at all in the bullet weights I use. I use 150's on WT deer and 165's for bigger animals.
To me, when I hear exaggerated differences between the two cartridges, I kind of think it's like saying that an 8" blade in a knife fight is better than a 7 and 7/8" blade - every time. Truth is, there are a whole lot of other factors that will probably make more difference.
I know a lot of people sneer at what gunwriters say, but I do like John Barsness and I kind of like the way he sums up the .308:
"Sometimes they'll even say it isn't adequate. What??!! With 180 grain bullets it matches the the long standard .30-06 velocity of 2,700 fps, give or take a few fps. Load a 180 grain Nosler Partition (or Hornady INterlock, or Swift Sirocco, or whatever) to 2,700 and it will kill anything up to elk and kudu neatly out to at least 400 yards, if you can shoot halfway decently. I know this because I have done it."