Actually, SAAMI is only in charge of chamber specs for commercial .223 Remington. 5.56MM NATO specs are controlled by NATO.
Forster commercial .223 Remington headspace gauges are:
1.467" No Go
USGI Rifle, 5.56MM, M16, Gage, Headspace, Max Field Service (P/N 7799734) is:
As you can see the military field reject headspace for 5.56MM NATO is considerably longer than commercial .223 Rem No Go (.006")
I have had the opportunity to check headspace on several new factory assembled Colt AR15A2 types with factory 5.56MM NATO chambers, and they all headspaced within commercial .223 Rem specs (bolts did not close on a .223 Rem No Go gauge.) The Colt owner's manual says suitable ammunition is ".223 Remington (5.56mm)."
Based on this, it should be safe to shoot either type of ammo in a chamber headspaced to commercial SAAMI .223 Rem specs- WITH RESPECT TO HEADSPACE ONLY.
A much larger issue, as BDH alluded to, is the length of the throat. Back when SAAMI first standardized the .223 Rem chamber, the cartridge was mostly only being used in varmint rifles shooting very short, light bullets, and SAAMI standardized on a very short throat. A 5.56MM NATO chamber has a very, very long throat.
If you have a .223 Rem minimum chamber with a short throat and you attempt to use ammo with long heavy bullets, they may jam into the rifling and cause an unsafe high chamber pressure.
A true 5.56MM NATO chamber with a long throat will shoot anything (in a milspec AR that is, not necessarily in a bolt gun...
In an attempt to improve accuracy, many manufacturers have come up with custom chamber reamers with throats that fall somewhere in between the extremes, such as the Wylde chamber BDH mentioned.
The bottom line is, you MUST know what chamber and throat you have in order to know what is safe to shoot... Deal with a reputable AR builder that will tell you exactly what chamber they use. Brand X barrels may not even be marked correctly. If in doubt, send your barrel to a knowledgeable AR smith and have the chamber and throat measured.
Here is an excellent article by Glenn Zediker that explains it way better than I can: http://www.zediker.com/articles/articles.html#AR_topics
Download the "AR Chambers" article... In fact you should read all of them while you're there.
Hope this helps.