As has been said, loading for military semi-autos, both 150grn class bullets ('06, M2 Ball = 150grn while 7.62, M80 Ball = 147grn) are loaded to 2,820fps so there is no real difference as they use the same powder as they both to varying degrees are sensitive to excess pressure on the op rods, however, because the '06'd case is bigger, the case density is less in the '06 which is why many CMP shooters say the 7.62 is more accurate. But, with heavier military bullets like the 173 grn class, the '06's larger case gives it an advantage as you can load it a bit faster.
In commercial hunting ammo for bolt guns, the '06 can be loaded with a larger charge of slower burning powder (Win 760 vs Win 748) which gives it a velocity advantage of a about 100-200fps (depending upon bullet type) with a 150grn bullet. Further, as the Win 760 charge weight is higher, the case density is higher and the combination of higher velocities and higher case density make for the potential of a very accurate long range round. Finally, Hodgdon recommends the slower burning Win 760 rather than Win 748 for bullet weights of 180 and 200grn.