Actually, the problem isn't with the cases, but the chambers that hold the case.
The .223 Remington chambers are usually rated for lower pressures and have tighter neck tolerances. So, firing a 5.56mm round in a chamber meant for .223 could result in a failure due to the case sticking or because it's not up to the task of containing the pressures.
Also, the 5.56mm has a longer headspace (leade) than the .223. This means that the bolt may not fully lock when a 5.56mm is fired in a .223 rifle.
Remember, you can fire a .223 round in a rifle meant for 5.56mm, but you should not fire 5.56mm in a rifle meant for .223.
Spot on.