Oxford is mentioned in LSE's undergraduate prospectus, Cambridge isn't, it's as simple as that, really. Surely, neither Oxford or Cambridge can be objective in their judgments of their relative brilliance, so we need a third party, which is where LSE conveniently enough steps in as our guiding hero and proclaims Oxford as the best university. Here is the direct quote from the prospectus: "If you prefer ... the traditions of dreaming spires, then LSE will not suit you."
On a more serious note, they are both brilliant academic institutions, and I don't think that on average the students at either university are better than at the other. However, I still believe that there may be differences for individuals; one university might have a particurarly good lecturer in a subject, the course offered might be the one best suited to the student etc.