Hi, I am in the exact same situation here, deciding between Cambridge Part III and Oxford MSc Mathematical and Theoretical Physics (plus I also hold an offer for Part III at St. John's!). I did not do my undergrad at either of the universities, but just by looking into the online notes and handbooks of both courses, I have the following summary which may be helpful:
Reputation and graduate prospectsThe Part III is more famous due to its history, this of course has nothing to do with its teaching quality, course structure, etc., but it may come in handy when applying for PhD/jobs in the future. One thing to note is that passing Part III (and quite often, obtaining a Distinction) is a
hard prerequisite for doing a PhD at DAMTP (in most areas, though it is definitely the case for the high energy and gravity groups), I believe some very selective groups (e.g. HEP and GR) even make Part III students take admission exams for their PhD in January, so if you want to do a PhD at Cambridge, or at least have that option open, Part III is the way to go.
Course structure and examsThe Oxford MSc has courses and assessments more spread out across their three terms. Specifically, there are assessments by exams and mini-projects towards the start of Hilary and throughout Trinity (Oxford names for Spring and Summer terms), some lectures are also entirely/partly assessed by coursework. On the other hand, Part III has most, if not all lectures in Michaelmas (Autumn) and Lent (Spring), then a big exam week in Easter, which is what your whole degree classification will be based on (with the exception of the essay, which may be offered as a substitution for one 3-hour exam). The Part III exams are also notorious for its difficulty, and personally I see the Oxford MSc as more forgiving/manageable.
Scope and breadthThe Cambridge Part III focuses more on mathematical physics, by this I mean even though they offer theoretical physics courses with the same name (e.g. QFT, GR), the notes seem to go into much more detail in terms of mathematical consistency and (sometimes) rigour. If you are studying specifically for theoretical physics, the Oxford MSc seems to offer a wider range of options, where without dwelling too much on mathematical intricacies, you'll be able to explore more topics, ranging anywhere from, say, geophysical fluid dynamics to superstring theory.
Of course, these are just my opinion based on things I found online, so take it with a grain of salt. Ultimately, your decision will also depend on more specific circumstances such as scholarships/funding, potential PhD supervisors and research areas, personal preference for the city of Oxford/Cambridge, etc. I am attaching some links below which you might find useful:
Btw feel free to send me a DM for further discussion as I too have not make up my mind yet
