When something is 5.56 mm x 45 mm it meets specifications demanded by NATO. For example, the brass is a touch thicker, primers are harder, primers are crimped into the primer pocket (a pain for reloaders) and there is a laquer seal around the primer to prevent water from getting into the case. Also, depending on which version you obtain, it's either going to be 55 g or 62 g bullets.
Manufacturers of civilian (.223 Remington) ammunition don't have to meet these standards. They have others.
I believe that automatic weapons and semi automatic weapons are going to have slightly larger chambers than those that aren't. That extra bit of play makes them more reliable. I think that a 5.56 chamber is slightly larger than one that is .223. You should be able to fire .223 out of the 5.56 mm rifle, but could have a problem the other way around.