I have been living in Indianapolis for 2 years.
Like KHughes said, Indianapolis has a truly thriving downtown. Its pretty amazing really. A mall of dramatic european-style mall of monuments. Really nice, first class museums (Indiana History, Eiteljorg Western Art, NCAA Headquarters and Hall of Fame). A quaint downtown zoo. Large urban park along the White River with concert venue. A park-like canal. Spiffing new sports arenas (Conseco, Lucas Oil, Victory Field). A very nice downtown mall as well as other shopping with grocery stores (e.g. Mass Ave.). A relevant Symphony Orchestra and several theaters. Indy is convention destination and hosts many National events (Final Four this year, Super Bowl 2012, Big Ten Tourney annually, and is a candidate city for the US World Cup bid 2018). And lots of people actually living downtown. People are out on the street every night. And overall the downtown is relatively densely packed together.
On the immediate outskirts of downtown things become pretty ghetto with a few exceptions of urban revitalization in progress (Fountain Square, Fall Creek). The city then sprawls out much like Columbus. There are some very nice older neighborhoods along the winding White River to the north in a location roughly analogous to Clintonville. (Butler-Tarkington, Meridian-Kessler). Broad Ripple in particular is a gem of bungalow homes, local shops, and college dive bars with a real sense of community. Then, like Columbus, the Dublin/New Albany like suburbs are to the north (Zionville, Carmel).
Columbus at this point has an empty, parking lot shell of a downtown that rolls up at 5 pm, with islands of revitalization in progress (e.g. Gay St.). Also in contrast to Indy, Columbus's strength is the neighborhoods immediately surrounding downtown (Short North, Victorian Village, German Village, Grandview). Plus Columbus has a major public university.
Basically Indy has a grown up downtown that is maybe something like 20 years ahead of Columbus. But Indy went a different way than Columbus. It seems to me that Indy was evisioned, planned, and massively invested in. And now there are problems paying for it all.
I have the impression downtown Columbus is being taken back block by block, park by park, small project by small project, kinda like what I imagine happened in the Short North. At this point it is seems much more organic than what happened in Indy. How Columbus will ultimately create its downtown? I think it is still up in the air, but overall its pretty exciting.