Actually, the necks of the cases are the same. The 5.56 chamber has more leade, or freebore, to allow out of spec or damaged ammo easier feeding.
The higher pressure associated with firing 5.56 ammo is because it is loaded to military specifications, not SAAMI specs.
The AR15, whether chambered for .223 or 5.56, is quite strong, and can well shoot either ammo in either chambering, and this has been done for years by many people with no ill results, notwithstanding Bushmaster's warning (which is inconsistent with Armalite's advise, and others). I feel that Bushmaster touts this to promote their "5.56" chamber over that of other manufactuers.
SAAMI's warning does not specifically deal with use of 5.56 ammo in .223 chamberd AR15's. It deals with use of 5.56 ammo in the very weakest of .223 sporting arms that may be in use. And ALL of the AR15 rifle and barrel makers know that it is possible, no, very likely, that 5.56 ammo will be fired in their .223 barrels. If it were really a safety issue, all of them would make only "5.56" chambers.
SEMI-AUTO chambers are generally looser than bolt actions. This is necessary to allow smooth feeding and extraction. It is also generally advised in reloading that ammo for semiautos be full length resized, rather than just neck sized as is common by target/varmint shooters who reload.
This looser chamber is why headspace gauges may give erronius indications in ALL AR15's, whether .223 or 5.56 chambers. It is a semi auto, it is supposed to be looser.