22-250. It can do what the 223 can do, but the opposite isn't true. A custom barrel to fast twist the 22-250 is even better!
^ this to a degree. A fast twist 22-250 would be very hard to beat. But it entirely depends on what your intended use will be.
Occasional short range varmints; I think they are 6's but I'd favor the 250 for the explosive impacts.
High volume short range varmints; I'd go 223, unless you wanna rebarrel often, then go 250 for reasons stated above.
The same can be said for medium or long range, but you will need the faster twist in either. Most 223's will come with a faster twist, but a 250 would spank them twist being equal.
Bullet weight is a big player for the longer shots, a 55gr out of a standard 22-250 might barely poke into and stay on a wooden fence post at 800yds, while a 75 fired from a 223 will splatter on steel. The only reason I mention it is to make sure your bullet weight/twist matches the ranges you intend to shoot.
I choose 223 just because of the volume I shoot, I'd love a 22-250, but I'd of burned several of them up by now. But I never feel undergunned with it.