In use, there's not really that much difference in those two .22 centrefire calibres. The .223 is accurate, fast and some say "civilised". It's the biggest seller. You'll always be able to find factory ammo. The .22-250 is faster, accurate of course and hits hard. They say it wears out barrels quicker. Which I'm sure it does, but unless you are stuffing a mountain of ammo through it, that really won't be an issue.
Frankly, they both do the job excellently with really subtle differences and whichever you choose, it'll be fine. I'd spend more time thinking about which rifle suits me, then decide if it's offered in both calibres. Or if buying second hand, I'd always go for the better rifle, whichever calibre.