This issue comes up WAY too often.
There is no difference between 7.62x51 Nato and .308 Win., they are the same thing. Your mother is also your father's wife. Two different women? No.
A military 7.62 chamber is generally more generous than a commercial .308 chamber. Military 7.62 brass usually has a thicker web to withstand MG extraction, and the internal case capacity is a bit less. This last bit isonly of concern to reloaders.
If your .308 or 7.62 chambered rifle is in good nick with good headspace, it can use any factory 7.62 Nato/.308 ammo.