Cambridge has traditionally been ahead in science and mathematics (e.g., it's where Newton was a professor). On the other hand, Oxford until the 1800s was pretty much a seminary and theological college. As a non-Brit, I still get the impression that Oxford is where you would find the best philosophers and other humanities experts, while Cambridge is still slightly ahead in STEM

Hmm ... Cambridge had Russell and Wittgenstein, e.g., you have that bass-akwards wrt Philosophy departments.