The .223 is generally though of as a more universal round for the AR-15 because of its ability to be shot from 5.56 chambered rifles or pistols.
More formally known as the .223 Remingtom, this is a rifle cartridge manufactured in 1957, by Armalite, to serve as the round for the AR-15 prototype.
Advancements that came from the birth of this cartridge come from Bill Wylde, who is credited for creating the .223 Wylde chamber, which comes with a free bore. This allows 5.56 NATO rounds to be shot due to the accommodation of a higher amount of pressure in the barrel.
However, because the 5.56 round itself is bigger, its pressure output is greater, making a 5.56 round in a .223 chamber dangerous to shoot.
When discussing the differences in the both cartridges they are relatively the same until you get to the barrel.