There have been some pretty heated threads about this subject in the past. Do a search and read up.
That said; I own and shoot both. .223 is nice out to 300yds, past that, IMHO, it doesn't have enough for rock chucks or cyotes. Will it do it? Sure, but not always cleanly. JMHO. Rock chucks are hardy little critters. I also believe that the .22-250 tends to be a more accurate round in most guns. Just another opinon, right or wrong.
With my Rem 700classic 24" barrel and reloads, my .22-.250 will reach out to 400yds and be two inches low if zeroed at 2" high at 100yds. Rock chucks go poof at 400yds with V-Max reloads that are just under max. I would bet if I were a better shot, hits at 500 would be just as potent and attainable. I agree that the .22-250 give you at least 150 yards more than the .223. It will however heat up quickly. If you like to shoot alot of rounds very quickly, .223 is a better choice.
Edited to add:
My AR with iron sights is perfect for a fun and good day of rock chuck hunting, but when I want to get serious, out comes the .22-250. The AR with irons is perfect for the distance that I believe the .223 is affective out to. Past the 300yd mark, the .22-250 really shines. They are both going with me on vacation to our cabin!