I have an HK91 that will blow the primers out of their pockets with most 308win ammo. It's a dangerous situation because they will occasionally end up down in the trigger group and cause a slamfire. It behaves perfectly with any 7.62x51 ammo I have ever used in it.No, the chamber isn't so different that your commercial ammo is going to blow up your military surplus rifle (with the possible exception of rifles made around the turn of the 20th Century, some had surface hardened receivers).
I would add the caveat that as long as it will properly chamber. I have three rifles with 308win target chambers that won't chamber 7.62x51 no matter how big a hammer you use.No, NATO ammo isn't of any substantial difference compared to commercial ammo so putting it in your .308 isn't going to blow it up.
And RCBS will also sell you one set of dies for both .38 special and .357 magnum. That doesn't mean they are the same cartridge. Aside from thicker webbing, the milspec for 7.62x51 allows for a .002" greater length from the bottom of the neck to the base of the case than .308win. Most major manufacturers of 7.62x51 (IMI and Lake City are two that come to mind) will construct their ammo to the upper end of this spec to promote better accuracy. This will also cause this round to not chamber in some .308win rifles.If there is a difference between a 7.62 cartridge and a .308 cartridge then show me the 7.62 resizing die, there aren't any, there are only .308 Winchester resizing dies.