There's about a 100 fps difference in muzzle velocity for SAAMI barrel specs for both the .308 and .30-06 with 180-gr. bullets at their normal max pressures. Some folks with tight .308 Win. barrels shoot a 180-gr. bullet from factory ammo faster than a loose .30-06 barrel with factory ammo with 180's; a very normal situation.
The .308 Winchester was made a commercial cartidge during the development of what became the 7.62 NATO round by Winchester (I used to think it was Remington). And it was based on the .300 Savage case as shown in:
'Twas the use of the .308 Win. and 7.62 NATO in NRA and military rifle matches starting in 1963 that 3 years later, few .30-06 rifles were seen in competition when either one was allowed. In 1966, the NRA and military bullseye targets' scoring rings were reduced by about half for those used up through 600 yards; too darned many unbreakable ties were shot. 5 years after that, the scoring rings on the 800 through 1000 yard targets were reduced the same amount. Bullets up to 250 grains have been used with great success in matches from both cases.
Afield after game, there's not much difference. Both can shoot bullets heavier than 200 grains with great accuracy in the field, but the .308 Win. needs a longer throat to do so. Before the .30-06 became commercialized, the .30-40 Krag was a favorite on the bigger game animals and many moose and bear went down from a well placed shot from one.