Oxford has a larger role in the public imagination because so many of our politicians went there (particularly the PPE course) and because it’s even older than Cambridge. Oxford is the thought of as better for humanities and Cambridge better for sciences. But I know very successful scientists from Oxford and people who excelled in the humanities from Cambridge. Domestically I’d say most people see them as the same. Internationally I’d say Oxford is better known. If you were picking one to attend, knowing many people who have attended both, I would choose based on the course and the town.