The rounds are not normally interchangeable unless you have a 5.56mm NATO rifle. If your rifle is chambered for .223 REM, it's recommended that you do not fire 5.56mm NATO from it. A .223REM rifle will chamber a 5.56mm round no problem, and vice versa, but the .223 REM chamber isn't made for it.
Difference in brass
On a 5.56mm NATO round, there is usually a + sign with a circle around it. This is an obvious marking difference, and the .223 REM will probably be stamped as such near the primer. If there is any internal difference in either brass, I don't know...I don't reload it and I don't shoot .223 REM, only 5.56mm NATO.
Difference in Chamber
The difference between the .223 REM and 5.56mm NATO chambers are in the throat.This is the part of the barrel directly in front of the chamber where there is no rifling to allow space for the seated round. The throat in a .223 REM chamber is a tighter spec while the throat in the 5.56mm chamber is a little wider. We're talking hundredths and thousandths of inches, though. I don't know the exact numbers off the top of my head. But, the added space in the 5.56mm chamber compensates for the added pressure inherent in the 5.56mm NATO round. This is where you have problems chambering a 5.56mm round in a chamber designed for .223 REM: this can potentially create an over-pressure situation outside of the chamber's tolerance. I'm sure there are a bunch of people that have fired 5.56mm NATO in a rifle designed for .223 REM with no problems, but if it's against the manufacturer's recommendations and something goes wrong with the rifle (even if it's not related directly to the ammo) good luck getting them to pay for it. I've never seen a .223 REM rifle destroyed by using 5.56mm in it, but I would not be willing to change it with my eyes/face/hands/fingers, etc.
To me, 5.56mm NATO is like a .357 round, and .223 REM is like a .38 Special. The rounds can be interchanged due to the size, and a .38 revolver will probably tolerate .357 rounds for a while, but you don't want to be the one holding it if it ever fails.