At the risk of getting my butt handed to me, I will attempt a clear answer. The Rem. 223 and NATO 5.56 are two different rounds. The 5.56 is a military round and the .223 is a civilian found. The case in the 5.56 is made a little heavier in places and the case neck is not quite as long as the .223. This COULD cause the bullet of a .223 round to be forced into the rifling of a 5.56 on chambering and cause higher pressures on firing.
The .223 can be fired in a 5.56 and a 5.56 can be fired in a .223, but, except in cases of emergencies, I would avoid it.