Well, I have to go with the .223. The 22-250 will out shoot it in the extreme distances; however, very few shots are "way out there". Ammo is much cheaper in the .223; more available in more places; and is not as hard on the barrel as the 22-250.
For closer in - out to 150 yards you don't even need that much cartridge. The old .22 Long Rifle will cover the first 100 yards and the .22 magnum will take care of the last 50 yards. BUT, the bullets do tend to ricochet way too much for my taste.
The new .17's are giving a good showing of themselves and the bullets really do explode on impact. Then, I don't own one yet.