The use of of the ".223" term is essentially short for the term ".223 Rem". Some ammunition manufacturers will include the "Rem" while others will not. The 5.56x45 NATO round will appear to be the same size as the .223, though it is a more powerful round than the .223. Since your rifle is chambered for the higher pressured round, 5.56, the .223 (Rem) round will automatically be included as a round that is safe to fire in your rifle.
However, if you rifle was chambered for only .223 Rem, you would not be able to fire 5.56 due to the higher pressure.
Both the .223 and 5.56 round will serve your rifle well.