Deerhunter125,
You've chosen two excellent varmint cartridges and it's difficult to pare it down to one...only one. I have a .223 (Win Model 70 Featherweight) and a .22-250 (Rem 700 Varmint), and have had others chambered for the same cartridges. I use the .223 for days when I expect to carry the rifle for a purpose stroll, and the .22-250 for the occasions when I expect to sit quietly and snipe at careless varmints who emerge for lunch or to enjoy the sun. I've nailed coyotes with both, and they had no idea what hit them. The most dramatic (if I can use that adjective) was a coyote hit low in the chest with the .22-250. He flipped backward, down for the count, and barely twitched.
Since it isn't a question of which cartridge is sufficiently powerful or accurate, the choice has to be made on the basis of other factors. If you don't handload and you rely on factory ammunition, the .223 moves to the forefront. If you compare the price of a box of .223 and .22-250, you'll get more (cartridges) for your money with the .223 and bulk packs are available at extremely reasonable prices. The .223 is a very efficient .22 centerfire and I really don't think it gives up much to the .22-250, especially since I load my .22-250 down (my .22-250 delivers its best accuracy with 55-grain bullets at approx. 3400 fps). The .223 can be loaded up to 3100-3200 fps to compete with that, but I usually use 50-52-53 grain bullets in my .223.
There are models and actions (Sako, Kimber, Rem Model 799, etc.) that are proportioned for the .222/.223 family of cartridges, and they make a great deal of sense. The .22-250 uses the same action as the .308 family of cartridges. I'm not saying it's wasted weight or bulk for a varminter but, if weight is a consideration, that availability of a more streamlined action for the .223 may affect your choice.
In terms of accuracy potentials, it boils down to the individual rifle, and I'm not going to express an opinion on that because there are too many factors beyond these cartridge choices that affect accuracy. In terms of terminal ballistics, the .22-250 packs plenty of "whack" at extended distances, simply because it has a greater velocity.
If you were pondering either a .22 Hornet or a .222 Rem, the choice would be obvious (.222). Deciding between the .222 Rem and .22-250 for coyotes? The .22-250 would be (has been) my choice. However, between the .223 and the .22-250, other issues have to be taken into consideration - availability, economy, personal preference, models that appeal to you. If there's a .223 that you admire, I wouldn't change my choice on the basis of whatever additional velocity the .22-250 would offer. If your choices narrowed to a .223 heavy barrel and a .22-250 sporterweight, I'd lean toward the .223. If you weighed the prospect of .223 HB and a .22-250 HB, I'd lean to the .22-250 (especially if you handload) but, in the long term, the .22-250 will cost more in terms of brass and powder. That difference may or may not be significant to you. My favorite .22-250 (my fourth chambered for this cartridge) has digested over 4,400 rounds (98% of them were handloads). My favorite .223 (my third) has fired a less than 2,000 but I can tell you that I nourish it very affordably. Both cartridges are fun to handload, great on targets.
If you handload and expect to consistently operate at longer distances and you want every advantage, the .22-250 is a good choice. I know I probably haven't written anything more than food for thought. I wish this was a yes-no, black-white issue; it's not. There are too many gray areas for me to be more definitive, especially since I like both of these cartridges. My best assurance? If you choose the rifle or barreled action well, you won't disappointed no matter which of these cartridges you choose. Enjoy either (or both) thoroughly.

Its a toss up, I've shot both and enjoyed shooting them. My personal prefference is the .223. They both are good up to probably 400 yds but best to 300 yds.

Great post EJP... 2 great rounds! To add a little, personally I would use the .22-.250 in any bolt rifle... just a little faster and you will fall in love with it. At one time I was sick in love with a .220 Swift but alas, its tender barrel just wilted too fast. The .22-250 invoked similar love of speed with a love that will last. I will say however, that for shooting running coyotes, you can't beat the .223 out of an AR platform. I shoot a Bushmaster Varminter and it is almost unsportsmanlike to have the advantage that it gives you for thumping coyotes. Hand loads shoot half inch groups and it is fast enough to minimize lead and wind impact. Extremely low recoil allows your cross hairs to stay on the running target while you pull the trigger. You can actually get about 1 well placed shot per second and you can usually drop the critter before he covers 50 yards for sure. You just can't do that with a bolt rifle and try as you might to make that first shot count, you do occasioally miss a running coyote or pull up on multiple coyotes. If you don't shoot at running targets, this is not such an advantage though.

Sorry about that:

In my opinion I would go with the .22-250, just for the simple fact that it shoots flatter. Living in Ohio, I only use mine to shoot groundhogs so either cartridge would suffice, but I was raised on the .22-250. I have shot plenty of .223s and they are great, but I still go back to the 250. If you zero in each cartridge for 200yds with a 55gr bullet the .223 is about 1.5" high @ 100yds. and about 22" low @ 400yds.; the .22-250 will be about 1" high @ 100yds. and about 14" low @ 400yds. To me that's the main selling point when I am looking for a rifle that will be shooting groundhogs from 50-500 yards.

I grew up hunting with the 22-250. I killed my first deer with it and almost never put it down. It was insanely accurate and I felt confident in the rifle. While I never shot anything over 400 yards with it I feel I could have with no problems. The only .223 I have fired enough to give an opinion is the M16-A2. We shot from the 500 yard line everytime we qualified. With those iron sights I would not shoot anything other than paper (or someone shooting at me).

With a well made rifle topped with good optics I would not hesitate to shoot over 400 yards with either.

Neither is better, they are just different. 22-250 is a bit faster, 223 is a bit cheaper to shoot and ammo is more plentiful. Just a matter of preference.

I would go with .22/250, love mine

both have their place, but I would not trade my 22-250 for a .223. I love the Hornady v-max in 55gr it shoots wonderfully out of my gun.

22-250 GREAt Caliber awesome ballistics, faster, more punch, perfect for prarie dogs to deer..

.223 can reach out to 600 yards + with the proper load. NRA Highpower is dominated with the .223, most guys shoot 75-80grain bullets at the 600 yard line, but some people push the little .223 out to 1000 yards using 90grain Berger's or SMK with super hot hand loaads. Don't underestimate the .223. I'd take it over a .22-250 any day.

if your talkin energy, velocity, wind drift,bullet bullet drop, and how big the exit hole on an animal is, then the 22-250 wins hands down. my cousin has killed deer at 500 yards with one.

Has anyone ever tried killing wild pigs with a .22-250? I know it is small caliber to penetrate vitals. But with the accuracy and short range shots <100 yrds, a well placed bullet to the ear or eye area? I think it would drop them but wanted to ask if anyone has tried. I saw a 60 gr partition bullet , thinking that may penetrate better than a hollow point varmint round? Any thoughts?

55 grain bullet in a 223 is 3250 fps, 22-250 is 3850 fps.

i love the 223 nothing is beter, thats why the military uses it.