The external case and complete round dimensions are identical between the two as are the chambers other than throat length and leade angle/length (ie what together is often called 'freebore'). A longer freebore reduces chamber pressure. SAAMI .223 Rem chamber spec is for a very short 25 thou' throat length and a relatively sharp angle/short leade. So, if a 223 Rem rifle has a true SAAMI chamber - and few if any actually do - then 5.56 ammunition can feasibly produce excessive pressures as it is theoretically loaded for the longer NATO-spec chamber freebore. (Note - not headspace!)
In practice, mass manufacturers simply don't make rifles with such restrictive freebores. The simple test if you want to see what your 223 rifle is like in this respect is to use the Hornady OAL gauge / modified case and just about any suitable bullet in your chamber. Nine times out of ten (plus!) rifles, the resulting COAL with the bullet just off the lands is over the max SAAMI 2.26-inch cartridge length showing it has 'excessive freebore' by SAAMI standards. (Heavy bullets over 70gn are a different matter and should use custom chambers so they are seated relatively far out, the 80gn Sierra MatchKing loaded to 2.550-inch COAL for use in suitable long-freebore chambers in competition rifles for instance.)
For the ins and outs of the matter including strain-gauge pressure tests of firing various makes of 5.56 military-spec ammunition in a true SAAMI-compliant chamber, read:
5.56 vs .223 - What You Know May Be Wrong - LuckyGunner.com Labs
This includes a very informative table of various reamer makers' chamber specs for 223, 223 Wylde, 5.56 and others, but note there are other custom chambers with far greater freebores than any listed here, the PT&G 223 ISSF for 223 F-Class rifles utilising 90gn match bullets with 169-thou' total freebore length as an example.
Chamber specs aside, the current 5.56 NATO round is loaded with a relatively long 62gn FMJBT bullet which a SAAMI spec barrel with its one turn in 12-inch twist will not stabilise.