For purely coyote hunting
Id vote for the 22-250
if you plan to do some range shooting and predator Id go 223.
for lots of shooting the 223 will out last your 22-250 ( i am talking 1000s of rounds)and its cheaper to shoot
if your just going to take the gun and go shoot a coyote may as well go for the faster 22-250
if your adding seals go with 223 . The 22-250 from my experience is too much gun for seals from a cost and damage standpoint. Having said that availability of cheap 223 ammo is not what it was. And with price of pelts the damage is no big deal but for open water hunting I have sunk more seals shot by 22-250 than by 223.
Personally I think the 223 is the way to go if your going to own one 22 centrefire. But for coyote only Id chose a 22-250
I'd go with the .223 - cheaper ammo, and it'll do everything you need a "varmint" cartridge to do. The .22-250 may be a faster round, but there isn't a coyote on Earth that's going to duck a .223 coming at him, so it's really immaterial.
You cant go wrong, there is no wrong decision.Both the bullets are more than capable for coyotes. Both are inherently accurate rounds.
Rule of thumb for 223 versus 22-250 is called the 20% rule. Its 20% more cost wise to shoot a 22-250, it will shoot 20% faster,drop 20% less and hit 20% harder
and probably wear out more than 20% faster.
Like i say its just a rule of thumb.... but not out by much.
Christopher is right when hit both will produce a dead dog.