As mentioned, the 5.56 and 223 cases can have differences in thickness and volume.
Some are the same.
I'm not sure when this rumor is ever going to die. .308 Win and 7.62mm NATO brass is substantially different as the 7.62 NATO brass was made with a thicker web to better withstand the extraction forces in machine guns with the longer .7.62mm NATO chamber.
The 7.62mm NATO chamber is .0013" longer than the .308 Win chamber as measured from the base of the case to the datum point on the shoulder. This aides feeding of the round in hot and or dirty chambers, but it results in more stretch in the case when it's fired due to the longer headspace, so a thicker web was warranted.
.223/5.56 Brass thickness and case volume
Unlike .308 Win and 7.62mm NATO, there isn't any difference between .223 Rem and 5.56mm NATO cases. With the possible exception of some reportedly out of spec .223/5.56mm cases made in eastern Europe there is no significant difference between .223 Remington and 5.56mm military cases.
As an example, I used to buy BHA white box (factory seconds) and blue box (remanufactured) ammo when I lived in SD. While their red box ammo uses new brass with BHA head stamps, their .223 Remington white box and blue box loads use once fired brass and if you look in those boxes you'll usually find a mix of military and commercial brass. However most of their loads are still easily MOA capable, despite the mixed head stamps.
In my own reloading, I also don't bother sorting cases in .223 and I still get 1 MOA or sub MOA accuracy in my .223 bolt guns, heavy barrel Match ARs and bull barrel varmint ARs.
In short, it just doesn't make a difference.
223 Rem versus 5.56 NATO chambers
As noted in posts above, the only difference between .223 Remington and 5.56mm NATO is the more generous lead and throat dimensions in the 5.56mm NATO chamber. The rest of the chamber dimensions are identical.
The Wylde chamber splits the difference, intended to offer a compromise between the shorter lead and potentially better accuracy of the SAAMI spec .223 Rem chamber and the more generous and pressure tolerant 5.56mm NATO chamber.
.223 Rem and 5.56mm NATO pressure differences
The other widespread rumor that abounds is that 5.56mm NATO ammo has a higher maximum average pressure standard than .223 Remington ammo. That isn't the case - The NATO and SAAMI standards are identical at 55,000 psi. The CIP standard adds some confusion as it is listed as 62,000 psi but it's actually the same pressure, just measured by a different method.
What does matter...
The longer throated 5.56mm NATO chamber does make a difference when 5.56mm NATO ammo is loaded to the 55,000 psi maximum average pressure spec in a 5.56mm NATO chamber.
If you shoot that ammo in the shorter throated .223 Remington chamber, you may encounter higher pressures.
Dangerously, higher pressures? Probably not. Most firearms are proofed to 125% of the maximum average pressure, so while you may accelerate wear you probably won't blow anything up if you accidentally fire 5.56mm rounds in your .223 rifle. I just wouldn't make a habit of it.
There's also a fair amount of variation from the blueprinted chamber specs. New chamber reamers cut larger chambers than worn reamers, so the rifle made with a brand new reamer will have a large chamber than the final rifle made with that same reamer. Consequently, your mileage will probably vary compared to other shooters.
Some ancient history regarding the pressure myth...
The myth that 5.56mm ammo has a higher pressure does have a gain of truth to it as back when 5.56mm NATO was new and the Earth had just barely cooled, the original US military M193 round had to penetrate an M1 steel helmet at 500 yards.
This requirement was achieved in the original prototype .222 Special cartridge (which was renamed the .223 Remington) using IMR-4475 and a Stoner/Sierra designed .55 gr FMJ bullet at the original specified maximum average pressure of 53,000 psi. This bullet a long ogive and had a comparatively high BC that allowed it to retain velocity well.
However, when Remington and Winchester started loading the round for military contract purposes, the military required the use of a Remington designed 55 gr FMJ bullet with a shorter ogive and a lower BC. The result was that this short and squatty bullet lost more velocity over the 500m range and it consequently needed a greater muzzle velocity around 3,300 fps to retain enough velocity at 500m to penetrate that poor sacrificial M1 steel pot.
This was aggravated by a specified shift in propellant to WC846 - a colloidal ball powder used in 7.62mm NATO M80 ball ammo. An example, of McNamara's accounting focused whiz kids in action. WC 846 could be made much faster (2 days versus 2 weeks), was much safer to make (it used a wet process), and as an added bonus could be made from WWII surplus cannon powder, making it much less expensive. However, this created a propellent problem as most lots of WC846 could not meet the higher velocity now needed with the higher drag bullet that had also been specified.
When Remington and Winchester both refused to make any ammo under the existing criteria (which they regarded as impossible to meet), it created a need for a variance on the maximum allowable pressure and the allowable average was bumped up from 52,000 psi to 53,000 psi to meet the requirement (and Federal wanted 54,000 psi). The original and rather wide WC846 powder specification was also split into two narrower specifications, WC844 and WC846, to allow the end of the specification that was most effective in 5.56 M193 ball ammo to be called WC844 and used for M193 ammo production.
However, given the current SAAMI and NATO specs (and the CIP equivalent) of 55,000 psi, that's all water under the bridge as 52,000 versus 53,000 psi is a moot issue.
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