If you're doing a lot of varminting at ranges inside of 300 yards (at a prairie dog town, for instance), the .223 is more practical. If you're shooting rather infrequently (say, on woodchucks or coyotes), and at longer ranges, the .22-250 is the better performer.
Basically, a "standard" load for the .223 is a 50- to 55-grain bullet at around 3,000 to 3,200 ft/sec. The same bullets from a .22-250 are at some 3,800 ft/sec. Roughly; I'm too lazy to go to the book.
, Art