I grew up in Columbus and now live in Cincinnati, OH. I've traveled to Phoenix for 4 days.
Personally, the Arizona heat in the summer is unbearable. This year we got into the 90s maybe twice this summer (global warming FTW). As for the winter; Cleveland gets hammered with lake effect snow but as you move progressively south the snow fall becomes less and less of an issue. I think we had 5 significant snowfalls last winter here in Cincinnati.
Weather aside, I think Ohio has some interesting qualities. You've got the biggest college town in the world in Columbus and two pro-sports cities on the north and south borders. Both Columbus and Cincinnati have world class library systems. The cost of living is reasonable with four bedroom homes easily available in the suburbs in the $150s. As with most major cities the public school systems of the actual city leave some room for improvement but the suburbs offer great schools and great communities. As for the economy, Columbus is relatively sheltered from major swings in the economy with both Ohio State and the State government holding jobs through rough times. Cincinnati is home to the Corporate Headquarters of 5 or 6 Fortune 500 companies including the behemoth Proctor & Gamble. I don't know as much about Cleveland.
Phoenix was a nice city with plenty to do but I don't know a tremendous amount more than the four or five major landmarks I hit. It all comes down to the climate for me.