According to the Speer manual, they are exactly the same round. The former is the civilian designation, and the latter is the NATO designation. Of course, the military used FMJ bullets while hunters typically do not.
GK,
Yes the .308 and the 762 are the same and yet not. The NATO round has thicker brass and the Military rifle has a deeper head space (this I got from an ammo MFG and reading on the WEB). You can fire both from either BUT the civilian .308 may over expand and jam in the military rifle.
Hazcat's assertion about the 7.62 having a slightly thicker brass case than it's .308 counterpart is correct, and is mentioned in my Speer manual No. 12, which recommends reducing powder charges by 5% just to be safe.
Fulton Armory has some expertise in this matter, and their comments are HERE.