It really depends on the terms of your hunting and the demands you place on your gun.

If you want a dedicated coyote rifle that also pops prairie dogs, then the .223rem is the piece of choice. If you want your dedicated coyote rifle to be able to take out deer as well, then get the .243win. They're both very capable coyote rifles.

As others have mentioned, the .223 tends to do less pelt damage, however the .243win, with the right loads can be made to do very little pelt damage...I typically have used deer type bullets for coyotes in the .243win, rather hard for coyotes, they do much less pelt damage than typical "varmint" bullets. In states where FMJ's are legal for coyote hunting, I'd recommend them, assuming you've got a good shot. You'll get a 6mm entry hole and 6mm exit wound.