I'm a Twin Cities native who now lives in Indy. You'll find many similarities between the two. Both places are very midwestern in their sensibilities. Not too flashy, somewhat family oriented, etc.
Both of the "a-polises" have thriving downtown areas.
Sports: Indianapolis
Obviously, Minneapolis has the Vikings, Wolves and Twins downtown. But, the Dome is a disadvantage, both in its location (disconnected from the rest of downtown), and for the teams that play there (especially the Twins. I'm excited for the new Twins Stadium-- I think it is going to be a great thing for the team and the Warehouse district area) As for Target Center and the Wolves-- it's not the building or location-- it's the team.
Indy is geared toward serving sporting events and conventions. The Colts, Pacers and Indians (AAA baseball) all play downtown in very nice, state of the art facilities. Indy also bills itself as the amateur sports capital of the country. The NCAA is headquarted here, and is why Indy gets the Final Four every five years. We also get Olympic trials here as well. The 2012 Super Bowl will be played at Lucas Oil Stadium. The Convention Center is going through an expansion. The RCA Dome was just torn down to make room for that.
When finished the Stadium, Convention Center and Circle Centre Mall will all be connected via a Skyway and tunnel system.
Downtown Indy caters toward these visitors. Circle Centre is a thriving shopping mall. There are tons of restaurants. Many of them are chains, and there are several steakhouses. Some give Indy flack for this-- but I think it's a matter of catering to its audience. Business travelers and sports fans like familiarity-- and this serves them. It's a nice, compact downtown that's easily walkable.
Culture: Minneapolis. (But Indy is charging hard)
I think downtown Minneapolis has more of a "big city cosmopolitan" feel. There is more in the way of music, renowned theatre (though Indy has a decent scene), local restaurants, etc. in the core downtown area. There's nothing like First Ave. or the Guthrie or the Walker Art Center in downtown Indy.
That said, Indy is really trying on the culture end. They are working on a "Cultural Trail" throughout six downtown districts-- that cater to pedestrians and bikers. Massachusetts Avenue and Fountain Square are thriving with small art galleries and theaters, and good independent restaurants. It has meant a growing base of people that are moving downtown to be right in the heart of it all. While not downtown, the Indianapolis Museum of Art attracts good crowds, has great exhibits and has a terrific Friday night outdoor movie program every summer.
Minneapolis tends to attract a certain type of educated, cultured, higher income populace in its city limits.
Indianapolis, population and area wise, is much larger than Mpls. Indy's city limits include ALL of Marion County. So you have everything, from downtown condos, to refurbished and gentrified older neighborhoods, rundown urban slums and suburban subdivisions all within the city limits. As a result, you have more of a socio-economic mix in Indy, due to the sheer physical size of the city.
Schools: Minneapolis
I think public schools in Minneapolis are far better than what you would find here. Indianapolis Public Schools is struggling with one of the worst dropout rates in the country. Contrast that with Minneapolis-- where high schools like Southwest and Washburn can compete with the better suburban schools.
Indy's suburbs, like in the Twin Cities, do have some terrific schools. Carmel, Fishers/Hamilton Southeastern and Washington Township (North Central) are the cream of the crop. If that is a primary concern for you, or private school is cost prohibitive-- I would advise living in one of those districts.
Outdoors: Minneapolis HANDS DOWN. Nothing compares to the lakes area in Minneapolis from a running, biking, beauty and people watching standpoint. Downtown Indy has White River State Park-- which is actually very nice. It includes the zoo, a canal in which you can run and bike around, or take a paddleboat out, as well as a really cool outdoor amphitheater that gets some good concerts. The Monon Trail, while not stunning from a natural beauty standpoint-- is a nice conduit to be able to bike from downtown up to Broad Ripple (think Uptown Mpls), and on to Carmel (think Edina or Eden Prairie).
Minnesota is a huge outdoors state with all the water and woods. Indiana just can't compare.
Workforce: Pretty similar. Life sciences is big in Indianapolis. Eli Lilly (the drug company) is headquartered here, as are insurers Anthem and Wellpoint, nd tech startup Angie's List. Lilly brings in A LOT of very educated people from all over the world. Obviously, the Twin Cities gets the same from 3M, Target, etc.
People/Customer Service: It's what you make of it. Downtown Indy caters to people in for the Colts game, national tournament or business convention-- so service is at a premium.
As for the people, it's the Midwest. In both cities, many people are from the area and have their established networks. So, it takes a little longer to break down those walls. People are very friendly here in Indy, and if you get involved in your neighborhood or community, you'll have no problem making friends.
Sorry this is so long! I hope this helps. Mpls. and Indpls. are both terrific cities. I'm glad I've lived in both. If I had to pick-- I'd probably pick Mpls., but only because it is "home".