Here is a link to Armalite explaining the difference.

http://www.armalite.com/library/techNotes/tnote45.htm

Here is another link to Fulton Armory.

http://www.fulton-armory.com/556-vs-223-Chambers.htm

As 7idl said NEVER shoot 5.56 Nato out of a .223 SAAMI spec barrel. If your ar15 is marked 5.56 Nato your fine you can shoot either one.