The difference is small, but it is there.
Sierra, for instance, sells its Varminter line in both .223 and .224. The fact that both sizes are selling is an indication that: a) there are a lot of gullible people out there or b) there's a difference in performance. Now, to be sure, both are correct, but I'd go with (b) if I had to make a decision.
Where are you looking for bullets?
No. They sell the two sizes because some of the older 22 center fire rifles were made with a .223 diameter bore rather than the industry standard .224 bore size that is common now. Mostly this shows up in 22 Hornets, but there are others. They shoot better with the correct size bullets. Sometimes. Some of the older guns shoot just fine with 'wrong' sized bullets. And to add to the joy, the .22 Savage High Power uses a .227"or .228" bullet. What happiness!
CZ still put .223" barrels on their Hornets, and as far as I know, nobody factory loads .223" dia bullets for them. It does not seem to bother CZ. Ruger made a bunch of their guns with barrels smaller than the standard diameter too, Mini30's with .308" barrels instead of .313 or so that the 7.62x39 is supposed to have.
Stevebot, the difference is .001" (one one thousandth of an inch), not .1 (one tenth of an inch).
So, to the OP, buy the .224 dia bullets to reload your .223 Remington cartridge. And do some reading through the various sources of info on cartridges, and you will find that there are huge variations in what the cartridge is called, vs. what size bullet it actually takes. Like the .303 British, which takes a .311 inch bullet, and almost all the 30 and 300 calibers taking .308" bullets (but not all of them, IIRC).
But your reloading manual should have that info. You have a reloading manual, right?
Cheers
Trev