5.56 uses a slightly larger bullet, as well has heavier weight. The charge is roughly 40% more volatile than .223 as well
Actions built for 5.56 are much more substantial and use military spec components and heavy springs. As well as cold forged chromoly barrels throats and bolts.
This isn't all accurate, at least for an AR15 platform. The difference in a .223 chambered rifle vs. a 5.56 chambered rifle is the chamber itself. Buy a .223 AR15, and put a 5.56 barrell on it, it'll fire fine, the components are all identical. The difference is in the cut of the chamber. 5.56 throats are not as tight, and the rifling engages later in the barrel. The reason for that is, if the same powder is used in a 5.56 cartridge, as a .223 cartridge with the same bullet, the 5.56 requires more powder to achieve the approximate 200fps faster muzzle velocity, thus less case capacity. .223 bullets when chambered tend to engage the rifling , while 5.56 do not. This extra resistance is not a problem with .223, but creates potentially dangerous pressure spikes, hence wider cut throat, and a few mm less rifling. If you look up pressure for both rounds they're max pressure is damn near identical because of these slight, but important chamber differences. As for bullet sizes, both calibers use the same .224 caliber projectile. As for different weights, the 5.56 NATO cartridges come in 2 different weights, XM855 is a 62 gn projectile while XM193 is a 55gn projectile.
Additional fun fact: Most commercial 5.56 barrels, and not cut properly. They're actually MUCH closer to .223 than you think, and if you get your barrel specked you'll see this. When I went through my armorer's school major name brand rifles needed to have their chambers reemed, and I saw the reason first hand. Too tight of chambers, lead to pressure spikes, and you suddenly have 5.56 cartridges losing primers into the action of your rifle. Out of spec barrels happened because chamber reamers wear down quickly, and their so expensive, manufacturers don't replace them quickly enough to garuntee spec chambers. PM me if you'd like to know company for certain that have this issue, and if you didn't spend 3k plus for your AR, it's probably out of spec. I will say this, we never saw a problem with Colt, they were all perfect . Probably because of military requirements.
Bonus fact: When firing a proper tactical bullet through ballistic jel, .223 tended to be MORE leathal than 5.56. The slower bullet expands more dramatically up close, causing more damage inside of a few hundred yards than the 5.56. The benefit to loading a cartridge to 5.56 velocities comes with super high ballistic coefficients, so they maintain velocity for longer distances, thus increasing lethal range. The XM855 does have a reasonably high BC, but who's really expecting to engage zombies at 600 yards with an AR platform.
My recomendation for a 223 cartridge: Anything with the Barnes TSX solid copper projectile. When we did ballistics tests in FBI certified jel, testing xm855, xm193, Federal tactical in 55gn and 62 gn, Hornady TAP (crappy useless horrible round, again message me for details), and Corbon TSX, the TSX Was the best, followed fairly reasonably by the Federal Tactical.