Given a fast twist (like 6.5 in a .223) and the right bullet (80-90gr VLD's), a .223 or .22-250 can be accurate out as far as 1000yds. Some intrepid souls even shoot AR Service Rifles in 1000yd competition (non-BR). *HOT* loads of N540 and JLK 90gr VLD's down a 20" 1-6.5 twist match barrel will get there easily. At 600 they don't give up much, if any, to the various .308 loads. And that should point out there there *are* better choices available, if you aren't constrained for some reason in terms of rifle or caliber.
Back to hunting, I grew up in western Nebraska, i.e the windy Wyobraska region, and spent an inordinate amount of time in any prairie dog town I could get permission for. Pretty much all of my shooting was w/ a .223 Rem (few thousand rounds) and a little w/ a .220 Swift (couple hundred). Not because the .223 is 'better' than the .220, or the .22-250, but because the bulk of the time, that was all I had.
The 'conventional wisdom' put the .223 as a 200-250yd varmint cartridge, and the .22-250 as a 350-400yd cartridge, and the Swift as a 400-450yd contender. Note that this is w/ light bullets, in the 40-55gr range.
In reality, it really depends, IMHO, on the conditions and the ability of the shooter. I'd say the values above are a good safe bet. I know for a fact that I pegged several prairie dogs at a shade over 400yds w/ 52gr TNT's. The flip side of that story is that it took me more than a few rounds to connect, and it wasn't terribly devastating (no 'red mist'). Out to 300 wasn't too difficult, and 350 was pretty doable. I never really got a chance to 'go for distance' w/ the .220 Swift, as the particular towns I had access to pretty much maxed out at about 450yds. It would easily do that far, and I took a few rockchucks here in Washington state at around 400yds w/ it, so it's good *at least* that far. From what I've heard, the .22-250 is in reality good out to 400+, and the Swift, if loaded hot enough, should be good to maybe 500yds.
I would personally rather use varmint bullets than the heavy (i.e. 80 plus grains) target projectiles on light/thin-skinned critters, but that's my own personal perspective. I'm not convinced that a bullet designed to survive being shoved down an 8, 7, or 6.5 twist barrel is going to do more than just perforate the target on contact. But that's my view. YMMV.
Monte