The .223 Remington and US 5.56mm ammunition are so similar that the cartridge cases would appear to be interchangeable. The specifications for the cases and the loaded ammunition are subtly, yet distinctly different, however. Although subtle, they may be enough to cause a severe problem, injury or death. This matter should never be taken frivolously nor should it be disregarded. If your weapon is chambered for 5.56mm, then you are likely good to go with either cartridge, however, if your weapon is chambered for .223, then you may indeed run risks if you fire the 5.56mm round, so don't do it. More here.
A .223 is not the same as a NATO 5.56. 5.56 weapons can safely fire .223 rounds; but, a weapon manufactured specifically for .223 cannot necessarily safely fire 5.56 rounds. Trying that can result in an unpleasant surprise for the shooter.