Covered before on here but briefly these programs are basically opposite each other in the Chicago area (although both have fascinating histories - Rush manufacturing the diagnosis of DID and UofC with guiding their psych program into a glacier).
Rush is a partially private practice based model place, low resident numbers per class, high volume clinical care (with lots of call), less focus on education and more about billing and realities of modern medicine. Their facilities are fairly nice, Gero unit which is either a big plus or big minus for most people (calls for falls at night on call), lots of consults encouraged on the floor to drum up more billing. They are not really a national name anymore (orthopedics notwithstanding) but pretty well known in the Chicagoland area.
UofC is obviously the name of a top 10 US university, that can never be taken away from them despite all their historic problems in psychiatry. They have tried their best to keep things Ivory Tower, heavy focus on didactics, they see less patients and do less clinical care. I hear psychotherapy training is pretty good. I think I heard a Rush grad comparing their caseload to a UofC and it was nearly 2:1 in total visits at the end of residency. Lots of driving to different locations in the city, location in Hyde Park is generally a bit of a pain for younger residents but there are great transport options from the loop. They have recently been rebuilding and medical students seem to have very favorable impressions talking to the folks there now. I suspect their resident class will get more and more competitive with years to come when it comes to STEP scores, grades, and publications if for nothing other than people's love to go to name brand locations.