Overall, I think the two towns are pretty similar: very close on the metro and city populations, similar distribution of wealth in similar parts of town, very similar in the landscape. As for education, Columbus does have OSU but I think that Indy's close proximity to both IU and Purdue allow it to compete for this category. To make distinctions, I would give the following as advantages for each.
Indy:
-MUCH better downtown. Some years back, Indy made it a point to invest in downtown and that has resulted in a lot of great facilities including museums, sports, restaurants, shopping, etc. Columbus has begun to develop the Arena district area, but has a long way to go IF they choose to try to catch up to Indy in this respect. And they may not, Columbus has other neighborhoods that kind of serve this purpose.
-More nationally recognized, partially thanks to the 500, Colts, and Pacers
-About 3 hours closer to Chicago if you're into that sort of thing
Columbus:
-The highway system is set up much better for access around the city and the speed limits are 65 through most of town.
-MUCH better neighborhoods that are CLOSE to downtown (ie Bexley, Grandview, short north, german village, Upper Arlington). Unfortunately you have to get a good ways out of downtown in Indy before you get to these types of places (ie Meridian-Kessler and points futher north). They are there though, with some beautiful/historic homes to boot
-Metro Parks. I think Columbus has one of the best Metro Parks systems in the country. There are so many of them and the ones I have visited are beautiful(ie Inniswood, Highbanks, Blendon Woods, Darby). Indy has Eagle Creek in town and Brown County not far away, but I have to give this one to Columbus.
As a side note, I've noticed some comments on some of these threads that are, to say the least, fairly asinine, biased, and misleading. This is just a friendly reminder to those who are here truly looking for information to make a decision, that it may be best to look at the concrete aspects of the cities discussed first and then visit these areas before making your decisions about the people or belief systems there. Although fairly conservative, both Indy and Columbus are good sized cities. All types of attitudes/beliefs can be found and I have found that either city has sufficient options available to make of it what you wish.