Remington developed the cases for 5.56×45 during development of what eventually became the M16. When the military decided to adopt it as M193 ball, Remington knew there'd be a lot of surplus brass and ammo around, so they took that identical design and designated it .223 Remington for the civilian market. At that point there was no difference between the two cartridges, not even as to pressure, as both were designated at 52,000 CUP maximum (though the military tech manuals called it 52,000 psi back then, it was actually as measured on an M11 copper crusher, so it would have been CUP in SAAMI lingo). Later on, SS109 ammunition was developed in Belgium and it measured a higher 55,000 CUP on the M11. The civilian pressures were not increased to match, though that's a smaller pressure difference than SAAMI allows for MEV (Maximum Extreme Variation of pressure in a ten round sample), so it's actually still perfectly safe and well below proof pressures.
The throat differences and slightly (0.001") wider chamber base diameters common in 5.56 chambers were originally about allowing long specialty bullets like tracers and easy full auto feeding in the M16 and M4. A .223 Remington chamber may see an increase in pressure with the specialty bullets, therefore, and will definitely see about 6% increase in maximum loads of M855 (the U.S. version of SS109). But most of the guns chambered for .223 that are not AR's use the same receiver and barrel and bolt lugs that are used in larger, higher pressure cartridges like the .308 Winchester, and so are not vulnerable to the slightly higher pressure from that round. A specialty round is another matter, however, and you want to check that tracers and incendiaries and other long bullet rounds don't jam into the lands of a .223 chamber as that can raise pressure upwards of 20%. Not quite to proof levels, but high enough to get primer leaks, widening primer pockets, and faster throat erosion.
Throughout all this, the external dimensions of the cases have remained the same for both 5.56×45 and its civilian derivative, the .223 Remington. For that reason, the same resizing dies work with both, with one exception: sometimes the once-fired surplus brass will have expanded and been stretched so much by full-auto firing and extraction under remaining bore pressure that a standard .223 resizing die cannot quite make it small enough for a standard chamber on the first pass. Small base dies are made to correct this. However, on a bolt gun, if you can force it into the chamber and fire it once, it will behave OK with the standard resizing die thereafter. Once in awhile you hear of an AR that won't reliably feed ammunition using cases that have not be resized in a small base die, but that is rare. Just be aware it is possible, so that if you get one of these guns you know what to do about it.