here's the scoop as I understand it.
5.56 is for semi-autos. These feed and cycle a bit better with slightly hotter loads, but also, the bullet flys free a tiny bit before entering the barrel, akin to a revolver barrel/cylinder gap. This means accuracy is slightly less good that what it could be.
.223 is the same ammo but tweaked for bolt action and single shot rifles, where accuracy is going to be more important, and as you hand cycle, gas for cylcing is of course not a concern. Also, these are sporting arms not war arms so rejacking a round and fiddling around aren't going to cost you your life, just maybe your trophy game animal. Anyways, because of these different concerns (accuracy vs reliabilty in a semi-auto) the bullet in a .223 chamber does NOT travel as far before entering the barrel. It's got less room. So if it was loaded the same as the 5.56 (designed for the more spacious chamber) in the 'cramped quarters' it would build up more pressure than intended. So they slightly downloaded the .223 to more meet the requirements of a boltaction rifle.
However, most boltactions you can get in .223 are also available in .220 swift, .22-250, etc, which work at higher pressures, so really you should be fine firing 5.56 in a quality boltgun.
For any semiauto, even if it says .223, it is probably designed with that extra spacious 'jump' for reliabity. The only exception to this are going to be certain heavy barreled varmint style semiautos tuned for the greatest accuracy.