In the years leading up to WW2 the Army wanted a new cartridge but with war fast coming, they stuck with the .30-06. Mainly because of the vast stock of cartridges and the fact that the M-1 rifle and the various .30 caliber machine guns were proven designs. Still, the fellers in the Ordnance Department were looking. The main reason was that the powder technology had improved between 1906 and the 1940s. This meant that they could use a more modern powder and get the same ballistics of the .30-06 in a smaller case.
A smaller case is a Big Deal to the Army. This means that there will be less cartridge brass used. It means that the loaded round will be smaller and lighter, and it means that the rifles and machine guns can be smaller. It also means that there will be a little less recoil.
Before anyone even thought of the .308, Savage brought out the .300 Savage, doing more or less the same thing, a smaller round giving about the same velocity as the original .30-06. This is a 150 grain bullet at around 2700 feet per second. This is a magic number for a rifle bullet. A spitzer (pointed) bullet at about that speed will hit about as far as a skilled marksman can aim. And a spitzer bullet with enough weight, say the 150 .30 caliber or the 154 8mm of the Germans can do everything an infantryman needs to do with a rifle. It can kill a man or a truck as far as the soldier can shoot. After than, call in the artillery.
So, by WW2 the .30-06 case had about a half inch or more of empty space that drove the Ordnance boys mad. Civilian hunters did not mind, the improvements in powder gave them more velocity. It was only fairly recently that civilian hunters learned that velocities above 2700 fps are wasted in about 99% of hunting situations, something the military learned back in 1906. This would be a separate issue but, offhand I'll just say that very high velocity is a special purpose affair, not a general hunting round.
Anyhow, the war wasn't over before the military started looking for a new cartridge. They wanted to keep the .30 caliber, after all, they had lots and lots of machinery to make .30 caliber barrels. They wanted the magic 2700 fps with a 150 grain bullet. I only know one name of the ordnance types involved with developing this round, Phil Sharpe. Philip Burdette Sharpe was one of the lions of the prewar handloading movement, writing one of the seminal books of smokeless powder reloading. This was A Complete Guide To Handloading, first published in 1937. As an aside, it's fun to read those old books, if for no other reason to see how folks took various things that today give the safety "experts" the vapors.
At any rate, somehow a bunch of Phil Shape's old notebooks came to the folks at Handloader Magazine, part of those notebooks was a long study of an obscure French Cartridge, the 7.5 mm. The .308 is very close to that cartridge case. Anyhow the .308, also known as the 7.62mm NATO is a pretty cool little round. It can easily beat the original 150gr. at 2700fps of the first .30-06. As a matter of fact several powders will beat 2700fps with a 175 gr. out of a 24 inch barrel.
Most .308s will not handle those long 220 grain round nose soft points, the one in twelve twist rates of the barrels will simply not stabilize them This is not a big deal, though. A lot of the newer, lighter bonded or all copper Barnes bullets will penetrate just as deeply. I would be very surprised if something like the 180 grain Hornady Interbond or that new Speer Deep Curl didn't do the job as well as, say, the old Remington 220 grain round nosed soft point out of the .30-06. Powder and bullet technology have advanced that far.
An open country hunter would be very well served with the Barnes 140 grain XBT slug, there are several loads that will give a muzzle velocity of over 2900 fps. Someone please explain to me exactly why I would need one of those real loud, real heavy Magnum rifles again. A hunter loaded with that bullet, at that velocity would have all the ooomph! needed to kill any deer, speed goat, or caribou way out past where 98% of all hunters should ever take a shot at unwounded game.
Something bigger out there? A 180 grain Norler Partition, Hornady Interbond, Swift A-Frame or Speer Deep Curl and you are up for Elk out to at least 250 yards. Moose? A little closer, although that's no problem, the only way to see a Moose at more than 250 yards is across a lake, Moose are not exactly plains animals.
With the exception of the Great Bears of North America, there is nothing but the Bison that I would hesitate to go after with the .308. To tell the truth, with a premium bullet, the Bison would fall but it just wouldn't be right. A Bison would deserve a Sharps or a Remington Rolling Block.