7.62x51 is the military counterpart to the .308 Winchester. The differences between the two involve headspace specs and pressure limits.
The headspace spec for 7.62x51 is more generous ( I don't remember the exact spec off the top of my head, but I think the difference is something like .002-.004").
The pressure limit for 7.62x51 is 50,000 psi. The pressure limit for .308 is 62,000 psi.
One final difference is that mil-spec ammo often has cartridge cases made of thicker brass than that found in civilian .308 ammo.
It is safe to shoot 7.62x51 ammo in either a mil-spec rifle or a civilian rifle with a .308 chamber. In theory, the potential problem comes when thinner cased, higher pressure .308 ammo is fired in a military chamber with a headspace dimension at or near the max acceptable spec for 7.62x51. In that situation there is more risk of a case head separation or case rupture, which could cause damage to the rifle and injury to the shooter.
Regardless of how they are marked, most rifles made for the civilian market are headspaced for the .308, so you are unlikely to encounter any problems firing either ammo type. However, you would be wise to check your headspace before firing civilian .308 in any military style rifle.
I am not sure what your source for the 3200 fps with a 149g bullet in a 7.62x51 is, but the information you have been given is wildly inaccurate. With a 147g FMJ, most 7.62x51 ammo will produce between 2650 and 2800 fps from a 22" barrel. A .308 can produce up to 2900 fps or so, but I don't think you will see that kind of velocity from factory ammo.
7.62x51 will not work well for hunting because it is loaded with FMJ bullets, which are illegal for hunting in most jurisdictions. If you somehow manage to get your hands on M118 sniper ammo, it is loaded with Sierra Matchking hollow points. That would make it legal for hunting, but they are still not ideal hunting bullets.
Your best bet is to buy a .308 of your choice, feed it with good hunting ammo, and go hunting. If long range stuff is in your plan, buy a good heavy barrel, bolt action .308 and hand load for it. I have a Remington 700 VS with the HS Precision stock. It is a bit heavy to haul around in comparison with your average sporter rifle, but it is very accurate.