I can give some advice on the non-academic elements (both are top engineering programs, so I don’t there is much difference there). I lived in both cities for undergrad and graduate school.
Location: CMU is in a fairly large city, although it has its own discrete, attractive campus in a very nice part of the city. PIttsburgh is a great city with lots to do and is a very easy, affordable city to live in. UM is located in a classic college town. Most students stay on the campus part of Ann Arbor, but there is another part to Ann Arbor. Most of what you would want is right on campus since it is huge. It is also spread out, with engineering classes being located on North Campus (requires a bus ride from Central campus, or you can live on North Campus). If you want the quintessential big American college experience, UM will offer that. If you want a bigger city, but more intimate college, CMU will offer that.
Atmosphere: UM has the big, American-style college feel. Sports (American football in particular) are huge; there is tremendous school spirit and a devoted (almost fanatic!) alumni base. CMU, being a smaller, private college, has a different feel. Sports are not as big, and while there is school spirit and loyal alumni, it is just different than going to a huge public University. Which you prefer depends on your personality. Both have very intellectual environments, with CMU perhaps being a little more so.
Weather: Both Pittsburgh and Ann Arbor have very similar weather. Not enough of a difference to really matter. Expect cold, fair amount of snow, from November to March.
Prestige: Both are very prestigious for engineering and overall. Any difference is not enough to be significant.
Students: both have large international populations. MIchigan, being a state school, has a much larger student population with a large percentage from Michigan. Both will have a fair number of wealthy students.
This is a tough choice since they are very similar except for size. I think it would come down to whether you want the All-American, rah-rah college experience. If you do, pick UM. If that does not matter to you or appeal to you, CMU may be a slightly better choice. I don’t think you can go wrong either way.