I have lived in Columbus and am about to relocate to Minneapolis, so I think I can give you a good perspective.
Minneapolis is a fun, vibrant, young city. It has an excellent municipal infrastructure, which means excellent public transportation (it's cool to "be green" and ride the bus there), fabulous public museums and art galleries, libraries, parks and trails, etc. It also is home to more corporate headquarters than Columbus is.
In my opinion, Ohio and Columbus are much more of a "hometown" feel than Minneapolis is. People there still haven't caught on to the benefits of urban revitalization (don't be caught downtown after dark!), public transportation, or how public funding for the arts and libraries can enhance the quality of life for those living in the city. It has been my impression that Ohio is still a little stuck in its ways in general, while Minnesota and especially Minneapolis are much more progressive.
I may be biased, but I think you will really like Minneapolis. If you're a young professional, there are tons of networking groups so that you can start to meet others in your same situation (young and relocated). The company I'm going to work for has about 60-65% of its employees from states other than Minnesota/Wisconsin/Iowa, and it's really supportive of people moving there. The company I worked for in Ohio was like, "Oh, you moved here? Why? Well, good luck meeting people and having a life since you're not from here."
Start checking out some of the websites below to meet people and see what's popular in each of those cities to get a better feel for yourself.
www.meetup.com
local.yahoo.com