If you had asked me yesterday morning, I would have said the 223 Rem & 223/556 NATO were the same. Originally, they were. The 223 cartridge was specially designed for military use. So the first difference is the primary use of the firearm. In the M16 (AR versions) the 223 was designed to fired from a 'hot' barrel, therefor the chambers were designed to reduce jamming. ... In civilian sports, the 223 is fired from a 'cold' barrel, so chamber design was tighten up to improve projectile accuracy. Mostly, this is about load ramp angle and distance to the lande. More jambs in the civilian sports firearm are only annoying, where a jamb in the M16 (AR) might be 'life & death' situation.
The next differences are the result of the military pushing the 223/556 NATO cartridge to its max. design pressures. Think 223 & '223 plus P'; and in resent years a new 223/556 with a heaver bullet was added for military use. This heaver round (slightly longer) loaded into the civilian firearm will contact the lande and push the light weight civilian barrels to their max. pressure or above. Quality civilian manufactures will state their firearms will fire the NATO ammo, but the life of the barrel will be reduced. There is a manufacture's liability here if an unknowing civilian loads the 223/556 NATO into his 223 sporter. ... Your choice ... anyway this is my overall understanding, and the basis of my statement that mil spec AR regardless whether the barrel is stamped 223 or 556 are designed for the 223/556 NATO ammo.