While its true that the .22-250 has greater powder capacity than a .223, and as a result can drive bullets of equal weight faster, that is only part of the story. A bullet has two velocities, it has a horizontal velocity and it has a rotational velocity. Over time the horizontal velocity decays due to the effects of friction with the air, but it's rotational velocity is affected much less by air friction, and at impact has slowed little from the speed it was rotating at, when it left the muzzle. Its uncommon to encounter a factory .22-250 rifle with a twist faster than 1:12, that is the bullet makes one revolution for each 12" of travel. A .223 that has a 1:7 twist will spin the bullet much faster, and so can stabilize longer bullets with higher ballistic coefficients, which will not stabilize in the typical .22-250. The fast twist .223 has been successful in 1000 yard matches.
Now, regardless of the game we shoot, bullet performance is the next most important consideration after bullet placement. The rotational velocity of the bullet is as important to the bullet's terminal performance on game as is it's horizontal velocity. A big game bullet for instance will penetrate deeper when spinning faster, because the precession (yaw) that the bullet experiences on impact is more rapidly recovered from with a higher spin velocity. A varmint bullet with a high rotational velocity becomes even more explosive as rotational velocity is increased, because the centrifugal forces tear the bullet apart with greater force a moment after impact. Even though a .223 bullet impacts with a lower horizontal velocity than the .22-250, the effect on the target, is also proportional to the spin velocity, just as it is to the impact velocity.