I have shot my 22-250 for a couple years and I had a .223 for a little while.
Cheap .223 ammo is about 2/3 the cost of cheap 22-250, the .223 has lower recoil just barely allowing you to see your hits. The .223 typically comes with a 1-9 allowing it to stabalize slightly heavier bullets than the 22-250s 1-12 or 1-14.
BUT, I love my 22-250. I handload 50gr BSTs and 60gr SPs for plinking and varmints.
I have personally shot coyote with both calibers and the difference is amazing. Coyotes hit in the chest with a .223 will usually do a bit of kicking n spinning or run maybe 20-50 yds. Coyotes hit in the chest with a 22-250 drop, instantly, period. It turns there organs to goo, seriously!
The 22-250 has an extra 100-150 yards of effective coyote killing range with similar bullets compared to the .223. The 22-250 is also capable of 4000 fps, very few cartriges can say so, only the 204 ruger comes to mind.
If you really want to shoot coyotes, and have a rifle built to do so, buy a 22-250. You wont regret it.
If you want to shoot lots of rounds cheaply, with the ability to hunt coyote, get a .223.
At the end of the day the .223 is a 300-350 yard coyote gun, the 22-250 is a 400-500 yard coyote gun. They both shoot lighter varmint bullets in factory ammo which make them vulnerable to wind deflection. If you dont shoot farther than 350 yards than the only difference is the extra punch of the 22-250.