Congratulations on your Cambridge offer. I think Oxford's MPP is more reputed and has more funding opportunities owing to the fact that it is an older course compared to its Cambridge alternative.
As far as what lies beyond the reputation of these policy schools, sharing solely from my experience as someone who has studied in the UK in the past, it is important to note that there are fewer opportunities in the UK for policy makers than in the US. Most of the policy organisations are either Brussels based or Geneva based with preferences given to people who can speak other European languages. The Oxbridge brand can get you interviews, but language plays an important role.