In my opinion, here are the differences btw CCF and Case:
Case - Very nice, down to Earth residents. The chairman, Dr. Nearman, is a very solid guy. Helpful, friendly, and funny. He is well liked and well known, at least in the Midwest(his name was brough up by the chair at Northwestern). The didactics seemed to be pretty good, the faculty are DEFINITELY invested in educating the residents. Most of them are really nice and helpful. University Hospitals is a nice, medium-sized institution. It is big enough to have some space (I hate the tiny feel of community hospitals), but it is not massive like its neighbor down the street. I feel like UH has a very smart group of people, not limited to the Anesthesia dept, who work hard and are friendly. Let's see, what have I left out? The dept takes care of its residents, lots of perks (so say the residents). I know that all of the residents I met while rotating there were very friendly and helpful.
CCF-Has a bad rep from the other Cleveland hospitals (UH and Case Metro). The other two hospitals see it as this massive corporate structure that bought out a lot of local hospitals and is slowly taking over the whole city. When you first see it, it is a bit intimidating. The Clinic is massive. It is also very modern and has some very beautiful hospital areas. In some ways, it does function like a well-oiled machine.
I found the Anesthesia residents at the Clinic to be very friendly. They are a smart group of people, but don't go out of there way to be all snooty like, because they are training at the Cleveland Clinic. They were actually very helpful to me during my rotation there. The Clinic does employ a lot of FMGs as a whole. I don't see this as a bad thing. All of the ones that I met in Internal Med and Anesthesia were very friendly and very, very intelligent. Most, if not all of them, were already physicians back home. In my opinion, they are an essential asset to training. They can help bring the interns up to speed on clinical practice, very quickly. So, basically I love the Clinic also.
I think that both offer excellent clinical training in friendly environments. The Clinic's name will carry you further, if you feel you need that sort of thing. Either way, you will be solidly prepared to be an excellent Anesthesiologist. I do, hate Cleveland, though, and not because of its weather. But that is also a personal decision for you guys to make.