Wrong. Slightly higher chamber pressure for 5.56 vs .223 ammo. Results in about 150-200 fps difference. Not a huge difference but some and specifications behind that nomenclature.
Yes, the thicker case for the 5.56 means pressure will be higher than a similar charge in .223. If you reload a max .223 charge in a 5.56 the pressure will high enough that you may get a case rupture.