22-250 for sure with one exception and that is if you ever intend to go on a prairie dog shoot. That's where I might give the nod to the 223. A 22-250 produces a lot more barrel heat and would shoot out a barrel faster when firing one shot after another for a long period of time. Otherwise the 22-250 will always outperform a 223.
Since you say you want a varmint rifle, I'm assuming you aren't wanting to shoot 100 grain bullets from a 22 in a desperate attempt to make it into a deer rifle. In that case, the fact that some 223's have a faster twist to stabilize longer bullets is not an issue. You don't want to pay for those bullets anyway if you are just shooting coyotes, crows, groundhogs, etc. Speaking of twist rate and bullet stabilization, one factor that surprises me when nobody ever considers that a 22-250 doesn't NEED as fast a twist as a 223 to stabilize a given bullet. When talking about stabilizing a bullet, the factor that matters is bullet spin rate not twist rate and you have to account for velocity to get bullet spin rate. For instance, a 22-250 will give you about 500fps more velocity with a 55 grain bullet compared to a 223. That's about 16% more velocity. With the same rifling twist rate then the 22-250 bullet would be spinning about 16% faster than the same bullet fired from a 223. This means you can stabilize the same bullet from a 22-250 with a 16% slower twist rate. Something to consider.
For me, the 22-250 is the ultimate factory varmint cartridge. Super flat shooting with 50 to 60 grain bullets with very light recoil and a reasonable barrel life as long as you don't shoot too fast to over heat it. A 55 grain Nosler BT loaded to the max can drop a coyote way out there.