While competition within their respective countries and number of institutions is a side-point, Edinburgh's international (and national) reputation irritates me (and oddly, particularly Edinburgh grads) a fair bit. In a way not-too dissimilar to the LSE, they've been accused of lacking feedback and contact with the senior staff, caring more about bums of seats (especially the ones that pay the big fees) than a students education, and looking to protect their image at all costs. This has manifested itself through closing departments that aren't performing well enough in research, even if they provide a valuable resource to students, focussing staff on output of cited work than teaching undergrads, and making money to buy in more staff for work unrelated to the original purpose of a university. Im sure much of what they do is very good, but it's upset a hell of a lot of students in the process. I've spoken to a couple of Edinburgh graduates who are on my program in Oxford, and they seemed amazed at how much the university seemed to care about supporting its students. There really shouldn't be such a marked difference in approach to student care as there is between two supposedly leading institutions, who try to set standards as high as possible.
McGill has big links with Scotland, and I've met a few of the students- they seem a happy and satisfied bunch with the way their university goes about its business, keeping costs to its students low at no advantage to the institution, and steadfastly refusing to grow to the size of Toronto (which it probably could). As a student, these are the things I care about, not whether someone on TSR might rate one above the other on reputation in their heads.