(I came across this 'debate' when looking for something else, and was intrigued to know the 'result' ... it seems there's no clear winner!)
They're both fabulous places. I'm a Cambridge graduate, and my daughter is still at Oxford. My personal opinion is that the stone which built the Cambridge colleges looks most magnificent in high summer, while Oxford stars in moody - more nuanced - light. If you really fancy having a go at punting on a river (active or chauffeured), with a stunning backdrop, Cambridge has the edge ... but you won't be the only one who knows that! If you arrive in winter, it seems there isn't much between Cambridge and the Urals to stop cold winds - a personal memory!
My best suggestion is that you choose whichever fits in with your other travel plans. Both cities/universities have strong links with the USA (my daughter spent time in "Dr Seuss's" old room!), and it's good to see you here!