It’s hard to predict which one will be better for you, but here’s what I can tell:
Few European schools have strong enough reputation among typical major MBA-employers in the US. LBS and INSEAD are clearly leaders there. I doubt that any of the other European schools would have more weight in the US than any of the American schools from the lower part of TOP 16 list (I mean Duke, Darden, Cornell etc.)
However, Oxford, and, to a lesser degree, Cambridge, could benefit not because of the reputation of business school per se, but because of the strong brand of the parent university. This mainly matters for general industry and international non-profit sectors – so, if your goals lie there, it indeed makes sense to consider Oxbridge (though for US, I believe Yale SOM would still be better).
Regarding this strength of the parent university name – in US, Oxford has better general recognition. Cambridge falls behind (maybe, because for US, the primal association with the name is Cambridge, Mass.) In the UK and Europe, however, both universities are valued equally. Well, you might meet opinions that Cambridge is better for maths, and Oxford for social sciences… these are mostly stereotypes and anyways don’t matter much.
Oxford and Cambridge MBAs are both well-rounded 1-year programmes, but their focuses are slightly different. Oxford’s strengths are social entrepreneurship and finance, and Cambridge’s is general management (and, to lesser extent, entrepreneurship and consulting). As for differences in schools culture – best advice would be to visit both places and decide for yourself, but I’d say that Cambridge ‘feels’ more personal and even a bit chaotic, and Oxford - more rational and business-like. Also (but maybe it’s just me), I felt that Cambridge is a trifle more closed and elitist than Oxford.
We had the discussion about Oxford vs Cambridge; you may check it here:Separating Oxford from Cambridge
Overall, if your goals were in traditional GM in the UK or Europe, I’d say that Cambridge would be better choice, and both would be equally good for non-profit. But for the US, it is not so – in fact, Oxford could give you more benefits there because of the perceived brand strength.
I wouldn’t count too much on the help from career services: in this market, these services at both schools are not that effective, and it seems they are targeted primarily at the UK market. Alumni network is also not that strong in the US because both schools are quite young (in future, however, we could expect that Oxford network will be stronger, because the school is more popular among North American applicants and, consequently, has a larger percentage of admits).
Sorry for such a vague advice – but I hope it still helped somehow.
Btw, are you considering some US schools, such as Yale?