I have two dedicated varmint rifles, a Winchester Model 70 Heavy Varmint .22/250 with 26" heavy barrel and a Browning A-Bolt varmint model with medium weight 24" barrel with BOSS. I have used both rifles for prairie dog shoots for many years. Although this is only a sample of two rifles and I am just some anonymous idiot on the internet, here is my experience:
My .223 is the most accurate rifle of any kind I have ever owned. If I do my part, it shoots cloverleaf groups at 100 yards. Nowadays I use mainly 40 grain Hornady V-Max bullets at about 3500 fps with a load about 1 grain under maximum with surplus AA2230C. The rifle is effective (by which I mean that I hit 75% or more of the prairie dogs I shoot at) to about 315 yards or so.
My .22/250 is a very accurate rifle also, but cloverleafs are rare...it shoots into about 1/2" pretty consistently. I use the Remington 55 grain Power Lokt HP bullet (purchased in bulk) with a maximum load of 40.0 grains of H414. It produces about 3600 fps and is effective (75% hits on PD's) to about 315 yards or so.
Both rifles produce identical trajectories with the loads I use. Theoretically the .22/250 with the heavier bullet will shoot flatter beyond 300 yards and buck the wind better...but I cannot tell any difference between them on live targets in the field. If the wind starts to howl or the range increases beyond 315 yards, my hit percentage drops with either rifle. If a prairie dog is hit with either rifle, it explodes just as spectacularly as it would if hit with the other.
The .22/250 uses more powder, more expensive brass and mine is a heavier rifle although that won't probably apply to you. The .223 is milder in report (I replaced the ported BOSS with the non-ported version to save my hearing) and recoil, is more accurate, shoots just as flat, doesn't heat up a barrel as fast, cleans a bit faster and costs less to shoot. It uses about 14 grains less powder per shot.
Of course if I used heavier bullets in the .223 or lighter ones in the .22/250, the larger cartridge would have a flatter trajectory, and the .22/250 has more power for larger game. But, if I was limited to one varmint rifle, I would keep the .223 and sell the .22/250 without hesitation.