I spend a lot of time varmint hunting with the 223 and the 22-250. To answer your question it really depends on how you will be using the rifles. 223 ammunition has went up a bunch in price, so pricing issues have become less of a factor.
I will tell you how I decide which rifle to hunt with each day I go.
I am a fixed position hunter so rifle weight is not a real issue with me.
I consider the 223 as a rifle capable of cleanly taking varmints out to about 300 or 350 yards(max). If you shoot varmints past that distance with the 223 then you are going to have "crawl offs". I have been able to extend this distance to about 450 yards with my Rock River AR15 Varminter. I have been able to extend this distance because I deliver a fast "double tap". That second shot will anchor them down. When shooting coyotes I really believe the effective "drop em dead" distance for the 223 is 300 yards. Any longer distance and you will definitely have "crawl offs" or "run offs" with the 223 cartridge.
Whenever I hunt a field that will have regular shots over 300 yards I take my 22-250(or larger cartridge). I feel extremely confident in my 22-250 out to 600 yards. That extra "ooooomph" is needed to anchor a varmint. Especially a larger coyote.
So to sum up what I do:
Short fast shooting out to 250 yards. I hunt with a suppressed Encore in 223 Remington. With a 40 grain V-Max bullet that is on a rope out to 250 yards. This rifle is short, lightweight, and quiet. Great for short distance.
When I set up in a field and shots are out to 450 yards max., I hunt with my RR AR15. the rifle is a heavy barrel. I shoot a 50 grain V-Max with the Leupold VH reticle. This is a great rig. I am going to add a suppressor to this rifle this winter.
When I hunt in fields out to 600 or 700 yards I hunt with my 22-250 and a 75 grain A-Max bullet. It bucks the wind well with a high BC. This is my tack driving rig. I am having another one built right now with a Krieger 1in8 twist barrel.
You really need to decide how you want to use your rifle. Then your decision should be simple. However if I was only going to have one rifle only. It would definitely be my 22-250. Tom.