Case students are actually quite collaborative, and also really love their school. So, I think the issues of engagement, collaboration, and school spirit are not especially relevant.
Personally, I would base this decision on cost and personal preferences, with cost taking the higher priority. CWRU and Rochester are very much peer institutions. Both have hospitals and med schools, for example. As a consequence both have a high percentage of pre-med students.
As suggested by others, CWRU has the better engineering school.
Rochester has a nicer campus, but Cleveland is the more interesting city.
I keep hearing that CWRU students are overloaded with work. UR is very demanding, but I know a student there, majoring in physics, and he does not complain about being overworked; only about how hard it is.