Over the years I have chronographed and accuracy tested a few different examples of .223's & .22-250's both in Rifles and Handguns. Generally speaking both cartridges shoot the same bullets, but you knew that. I have found that most of the time a .22-250 Rifle will produce 400 FPS more velocity than a .223 Rifle of similar barrel length when both cartridges are shooting the same weight bullets. Now in handguns that difference drops to 250 to 300 FPS difference between the two cartridges, again when both cartridges are shooting the same weight bullets.
The main difference is matching the bullets sectional density to the velocity to the intended game. To put it another way, a .22-250 Handgun (15" BBL) will normally shoot a 50gr. Bullet to about the same muzzle velocity as a .223 Rifle will shoot a 55gr. bullet. However I have tested loads in a Super 14 Contender that pushed a 40gr. Nosler Ballsitic Tip as fast as most .223 Rifles will shoot a 50gr. Bullet.
In my experience the .223 Handguns I have tested have been much more accurate than the .22-250 Handguns I have tested. Granted I have played with several more .223's than I have .22-250's. In fact the last .22-250 Handgun I worked with was a Savage Striker that belonged to a friend of mine. After shooting well over 200 rounds through this handgun we gave up. The absolute group we were able to obtain was 1 1/2" for 5 shots at 100 yards. Now that may be OK for a Deer Rig, but not for a Varminter in my opinion. I am not sure what the problem was, but I never was able to find the right combination for that particular Handgun.
Not really sure what to tell you, except maybe that with proper bullets and shot placement either will cleanly harvest Coyotes at reasonable range. The .223 will be a lot less expensive to shoot, so in the long run before you start handloading you'll be able to shoot more per $$$. The .223 is also not very intimidating to shoot and is quite mild mannered, not that the .22-250 was intimidating either.
Good luck with your decision.
Larry