Quoted: With all things being relatively equal whic cartridge should be more accurate at 300 yards. thanks
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Accuracy: The ability to place a fired round into the same point of impact as previous rounds fired at the same aiming point.
Precision: The ability to hit what you are aiming at, preferably with high levels of accuracy.
Given those parameters, I'd say this. Both cartridges will shoot exceedingly well at tht range; whether you or the rifle or ammunition you have are capable of that accuracy is a big variable. All things equal, the .22-250, by virtue of its flatter trajectory (muzzle velocities of 200-600 fps greater than a .223) with a given bullet weight and design is likely to give you an edge if you are shooting at an unknown distance and be slightly less suceptible to wind deflection.
I've shot .223 varmint rifle extensively, and can head shoot a groundhog at 400 yards with 7 minutes of elevation using 55 grain nosler ballistic tip at 3200 fps. My buddy, shooting a 50 grain NBT at 3800 fps, needs barely 5 moa elevation, and his gun shoot like a laser beam.