I'd say they're about even.
I was paying $745 for a one bedroom apartment in denver, and a decent one bedroom here goes for anywhere from $600 to $1000.
The cost of commuting is WAY less in Chicago, because you don't have to drive everywhere. Everything out is about the same, although dining out is cheaper here, near as I can tell.
Since Chicago is closer to the east coast, things that require shipping are generally cheaper, while other things, like real estate, are a bit more. But not much. Denver's gotten REAL expensive in the last decade or so.
Now, if you can live in Saudi Aurora, yeah, it's not too expensive, but I lived there a long time ago, and honestly, it's just too far outta town. I like living near downtown. In Chicago I live 15 miles from the Loop, 30 to 40 minutes by bus/L train and my 3 bedroom 1700 sq ft apartment is only $1100. In Denver, near downtown, in a decent section of town, my apartment would be nearly $2,000 a month.