Bullet weight in M1 is limited to under 180gr so no real benefit to '06. Marginal increase in velocity in some loadings, but for the most part .308 offers more versatile loads in factory ammo.
If you have other guns in .308 no need to stock or load a new caliber.
For me though, I'd pick the '06 for no good reason except that's how John C. built the thing, and US .30 M1 is what it's supposed to shoot.
A magnificent battle rifle still today, the long gun equivalent to Browning's 1911.
Correction, US .30 M2 is the 150gr military load designed for the Garand; .30 M1 is actually the 1926 machinegun cartridge.