If you handload, the '06 is superior in most loads except light bullets in carbine length barrels. Here it's so close I'd opt for the short action reliability of the .308 in say auto loaders or pumps.
Try this method to determine a cartridges inherent strength and additional advantage over the .308.
Determine the maximum effective range you consider a .308 good for on a particular game first.
Then look at velocity and energy at that distance that did the clean kill.
Next using the same bullet weight/type, checkout where on a ballistics chart, a 30-06 is pushing the same speed ,then a 300H&H, then 300wM, 308Norma, 300 Wby etc.
This is where additional powder and power comes into play, extending your max effective range. A clean kill at 100yds on an Elk with a .308 is a clean kill. Most '06s add on 50 -75 yards, and the magnums add about 150 yards to your barrel length.
For myself, I'll say my .308Winchester has a maximum effective range of 200yards on a large Moose. They can be killed much further, but I'll limit myself to the bullet performance of 200yards and call it the sweet spot and no more.
For apples to apples I'll take winchesters ballistic chart and use 180 gr Power Points in 4 popular .30cals.
1. .308Winchester @ 200yards we are down to 2176 fps and 1891 ft -lbs energy. Still fast enough for hydrostatic shock and enough energy to open up the big power point.
2. .30-06 we find the power point dropping to a similar speed and energy when it reaches out to 250yards, 2141 fps and 1831 ft lbs. hmmm.....a 50 yard gain as I expected.
3. .300WinchesterMagnum, we find Winchester's glory boy down to similar energy as the .308 at 200yards, when she reaches 400yards

, 2158fps, and 1860 ft lbs of energy remaining.
4. .300 WinchesterShortMagnum, we find her equaling a 200yard .308 hit when she gets to 400yards, 2166fps and 1874, a ballistic twin to her 50 year old sibling.
You can kill further with excellent marksmanship with any of the above cartridges on specified game,but we have to draw a line somewhere for a ethical comparison right.
Some think the magnums just make more noise,... and their standard calibers are actually a better killer. Of course this is absolute nonsense, tribal knowledge gained around campfires as a youth roasting weenies and baking potatoes.
If you use the above method I've used for years, you can take any cartridge in a given caliber, note it's optimum performance based on your own observations and where that performance starts to wane, and for any other cartridge in that diameter, check out where it falls with a similar weight bullet in velocity and energy, and you have an insight into its performance without ever firing a shot at an animal with it.
Remember it's apples to apples, notwithstanding bullet placement, bullet construction and not weight alone, twist rates, inherent accuracy and the list goes on, should be avoided, to determine a potential factor rating.