IN GENERAL, the .204 is flatter and faster than the 22-250. Some 22-250 handloads may be equal or better but MOST factory loads for the .204 perform ballistically better than MOST factory loads for the 22-250. Weather, wind, bullet weight, and final target may make a difference.
The .204 actually burns LESS powder than the 22-250 or .223 so it is actually quieter than either of those two guns, and, less muzzle jump and recoil.
Think of the .204 as a 22-250, only more pleasant to shoot!!
On to coyotes! .204 out to 250 - no problem. Past that, if the wind is light and you can put a bullet in the right place, it's still acceptable. Beyond THAT, the bullet gets a little light for my tastes and I'd prefer a .243/25-06 for shots over 300 yards and/or in strong wind. Incidentally, I have the same opinion about the 22-250.