I had to make that happy choice when it came to my masters degree, as I was lucky enough to get offers from both. I picked Oxford because it was generally regarded as the more prestigious of the two for that course of study.
For many courses only one university teaches it but not the other. For other courses one university is regarded as superior to the other. But for a huge swathe of subjects, it really is just a matter of personal choice.
Interestingly, when it came to my doctorate I was quite keen to go to Cambridge – really just to see somewhere new. But personal circumstances conspired against me, and I ended up going back to Oxford.
I prefer Cambridge, but of course I would.
The main differences I see are:
Cambridge is a town. Oxford is a city. Sure, Cambridge has a decent sized population and all that, but it feels like a town – it’s quiet most places at night and it’s mostly pubs not clubs, it’s just sleepier and less busy than Oxford. This might be a positive or negative point for you depending on what you like.
Cambridge is mostly the university and things that have come out of the university, whereas Oxford feels like it also has its own identity as a place. Again, you might prefer one or the other.
It depends a lot on your college, but on average Oxford seems more upper-class and unapproachable in places than Cambridge. Both have their dens of impenetrable upper-class tradition, but Oxford seems to take its much more seriously, whereas Cambridge mostly seems to understand it’s a bit silly and invite you in to have fun rather than mocking you for not knowing what you’re doing. (I come from a lower middle class background without a history of attending university.)
Similarly, Oxford takes a more traditional approach to some subjects than Cambridge does. I liked the way I could pick a variety of science courses alongside my first year of Computer Science at Cambridge.