I have a 22-250 and bought it because i wanted that specific chambering. It is an amazing round. Accurate, fast, flat, and powerful enough to take deer sized game if desired.
Unless you just want one or need the performance of the 22-250, you would be well off with the .223. It is a great round with alot of capabilities with predator and varmit sized game. Its no slouch either and ammo is so easily available and significantly cheaper if you plan on shooting often. With hunting predator sized game, its still easily a 300+ yard round if your set up is dialed in.
You will not have made a bad chioce with either but you should consider the type of rifle you want and your intended purposes, as stated above, when determining which will suit you best.
Infrequent use (hunting and occasional targets)
and hunting at long range- 22-250
Plinking/targeting alot, 100 and 200 yard hunting,
or just plain "fun gun"- .223