The 5.56x45 nato is loaded to higher chamber pressures than the .223 Rem, therefore you can shoot .223 in a gun chambered for 5.56, but not the other way around in semi-auto rifles, especially AR's. The .308 on the other hand is the exact opposite. There is a miniscule difference in case size between the commercial .308 and the 7.62x51 military rounds, the brass is thicker in the military brass, and the commercial ammo is loaded to higher chamber pressures. All these things combined "can" cause problems such as case rupture and head seperation in rifles with "fluted" chambers such as the CETME/HK clones. The higher pressures can cause other problems in other mil-surp auto rifles.
I hope this helps.
Don