To confuse (& clear up) a little more ....
Actually, .308 INCHES = 7.823
7.62 MM = .300 inches .... (do the math
Difference in the exact same designation (IIRC) is that the US system measures groove depth, while the metric measures to the lands (groove is the final cut maximum diameter of the bbl while the lands are the starting/minor diameter). Obviously, the groove depth will be a larger reading. So, all things being equal, you'll have a .004" groove depth (1/2 of the diameter difference).
This only deals with the bullet itself and has absolutely nothing to do with the configuration of the case itself - whole 'nother deal.