The simple answer is no.....but yes you can.
There are differences in chamber specs and maximum pressures. The SAMMI/CIP maximum pressure for the .308 Win cartridge is 62,000 psi, while the 7.62x51 max is 50,000 psi. Also, the head space is slightly different. The .308 Win "Go Gauge" is 1.630" vs. 1.635" for the 7.62x51. The .308's "No-Go" dimension is 1.634" vs. 1.6405" for a 7.62x51 "No Go" gauge.
That said, it is normally fine to shoot quality 7.62x51 NATO ammo in a gun chambered for the .308 Winchester (though not all NATO ammo is identical).
Nobody makes 7.62mm (NATO) ammo that isn't to the .308 "head space" dimension spec. So 7.62mm ammo fits nicely into .308 chambers, as a rule.
You CAN encounter problems going the other way, however. A commercial .308 Win round can exceed the max rated pressure for the 7.62x51. So, you should avoid putting full-power .308 Win rounds into military surplus rifles that have been designed for 50,000 psi max.
Should also note that military ammo often is made with a thicker web. Consequently the case capacity of 7.62x51 brass is usually less than that of commercial .308 brass. You may need to reduce recommended .308 Winchester loads by as much as 2 full grains, if you reload with military 7.62x51 brass, such as Lake City or IMI.
The Saiga 308 rifles are chambered for the following calibers: 308Win (7.62x51). The Saiga 308 has THREE locking lugs (all other 308 AKs only have 2) and was made to shoot high pressure hunting ammo from it's inception and incidentally seem to perform more reliably and group better with hot commercial ammo.