As has been said before, the cartridges are identical and fully interchangeable. 5.56mm is the military designation and .223 Remington is the civilian designation. The difference is in chamber dimensions, the military chamber has a longer throat than the civilian SAAMI chamber.
Many civilian chambers have the military throat dimensions and can use military spec. ammunition without problems. If you were to do the unthinkable and read the owners manual that came with your rifle the manufacturer will tell you if shooting military ammunition in it is a safety concern! But why would you want to read the documentation that cane with a gun???????