5.56 is just loaded to higher pressure. Its not advised to run 5.56 in a .223 chamber and here is why.
.223 chamber has tighter specs. for accuracy, what matters here is where the rifling starts and how much the bullet has to jump from the smooth chamber to the lands and grooves.
5.56 chamber has a greater distance.. so the bullet doesn't hit the rifling until its cleared some volume of space.. this allows for pressures to be controlled so you don't get a kaboom.
.223 the bullet travels less distance before hitting the rifling. Less volume means greater pressures. So .223 will have a slightly less charge with the understanding that chamber pressures will spike sooner.
There are many barrel types out there, but .223 specific are ussually chamber for match style barrels and accuracy, to only be used with .223 ammunition.