- The Maths Institute at Oxford doesn't offer MASt degrees and Cambridge doesn't give MSc's at all, they only seem to grant MASt's to external applicants who complete Part 3 of the Tripos without doing the rest of a Cambridge undergrad. At both of these universities these courses are taught Masters degrees.
They are the same in that they are taught programs, i.e., require a significant component of taking classes and passing exams. In my experience the precise letters after the M (Phil/Sc/ASt/etc.) don't really give great information at all about the content. You need to look at the details for each program instead.
The taught MSc's are nontrivial feeders into PhD work. As an anecdote: I got a taught MSc at Oxford, the program had a dissertation as a requirement in addition to taking courses, and I ended up with a DPhil offer there. Didn't manage to find the funding, so the PhD's from somewhere else.
- Their Masters by research programs like the MPhil have minimal taught components. That is to say that the only requirements are that you write a thesis, and you're not expected to take courses. I never met anyone doing these degrees or who has done one while I was there or since. They're so odd that Oxford's page for their version notes:
It is not intended as a first step towards a DPhil, but rather as an alternative to a DPhil. Few students opt to apply for the MSc by Research unless there are limitations on the time or funding available.
I simply can't recommend these types of degrees. In the US they're usually only given to students who start a PhD but don't finish all requirements for whatever reason.
If you don't have an undergraduate background in math or some equivalent experience I'm not sure that anywhere reputable would even take you on such a degree. But even if you do, suppose you finish a research masters. You've taken few advanced courses, and your main accomplishment is writing a "PhD lite" thesis over the course of 2 years.
A. If you decide not to continue on to a PhD, industry employers will be a lot less interested in a thesis than if you had just taken more advanced classes and built up a relevant skill set for their needs.
B. If you try to continue on to a PhD, in the best case you can't really extend your Masters thesis to accomplish that since it's already published. You're likely to have already sunk 2 years into the Masters and need to do another 3-4 on something different for a PhD.