There are a couple small differences in the two chambers.
The 5.56 chamber has a slightly longer leade to the rifling, and the neck area of the chamber is a bit larger in diameter.
This is because military spec brass is thicker than commercial, and it needs the extra room to expand and let go of the bullet.
Often, both rounds will seat in a 223 chamber. They will both fire in it. However, a pressure spike is created by the extra tightness as the bullet must be forced from the case neck, and then it hits the rifling sooner. This is also what leads to the slightly better accuracy of 223. The headspace mostly overlaps, but 223 goes more to the tighter end, and 5.56 goes further on the long end.
This pressure spike is not normally enough to blow up a gun, but it is an overpressure condition, and will cause accelerated wear and fatigue over time. In semiautos, it can cause the primer to be popped out as the action opens.
The wylde chamber was developed to get the best of both worlds. The tighter accuracy of the 223 chamber, without the pressure spike.