.308 is a shorter round, designed to allow a faster and shorter action for military semi and fully automatic weapons.
As a side benefit, it allows shorter receivers and actions on other types of rifles, too. So, a .308 bolt rifle can have a shorter receiver, slightly less weight, and slightly faster action (less throw required).
The .30-06 can accept heavier bullets (I have seen up to 220 grain) without exceeding OAL or pressure, and might be better for larger game.
The .270 offers a slightly flatter trajectory, and might be better if long distance shots are the norm.