I don't know much about the Cambridge one but I graduated from the Oxford ALSLA course.
Oxford's research in education is rated much better in REF.
Oxford's is the most competitive master's in the department by acceptance rate, and many of my classmates had already gotten a master's degree elsewhere (from Cambridge, SOAS, Edinburgh, HKIED, UCL etc). One of them had already gotten a PhD from SOAS and was a part-time university lecturer; one of them was working for the OUP, one of them was teaching at the Language Centre. Most were qualified teachers/had postgraduate qualifications and experience.
All but one were multilingual, there were people from the UK, the Netherlands, Turkey, Hong Kong, China, Taiwan, St Lucia, France. There was no-one from the US in my cohort but I believe that's unusual.
There's a departmental prize for the top scoring dissertation and it seems every year it's gone to an ALSLA candidate.
Oxford's department is smaller because it doesn't have undergrads.
Many people go into a DPhil afterwards.
Multiple professors teach/supervise on the course, including a former director.
It's an MSc by coursework (instead of research) but there are no MScs by research in the department ('political' reasons within the university) and it's research-focused (Trinity Term off) so you won't be disadvantaged. If you want to go into academia, the title 'applied linguistics' may be better for you.
You also get enrolled into a statistics course even though it's not evaluated.
Regardless of your eventual college membership, you are entitled to use the meal facilities at Lady Margaret Hall.
You get to see Richard Dawkins walk his dog next to the department from time to time.
Some graduates published their dissertations with an academic afterwards.
You get £100 from the department for research-related activities on top of whatever your college gives you (usually a lot more).
There's an event every year with the Cambridge candidates, presumably on the course you mentioned.
Oxford city is a more interesting city with more things to do and see, with tourists not too packed in one central area, and with a greater historical significance (first library room, first library, first public museum, first music room, first academic society, first botanical garden in the British Isles, oldest surviving purpose-built museum building, site of multiple parliaments etc).
Most people will go with a graduate college with many at Kellogg (my college). Kellogg has the best food and the cheapest ball. Second-largest college by student number. But many also were with a traditional college, eg Worcester, Regent's, St Hilda's, Jesus.
100% employment rate.
More (take-home) examinations (papers/assignments) then Cambridge I believe.
You also get a free English language teaching course and you can get an additional TEFL certificate.
Just went on Cambridge's website. On first glance I can see that:
Oxford's a bigger course.
Oxford has more star academics (no professors on the Cambridge one).
Oxford had more home students (32% v 14%).
Oxford has four more modules (doubles the content and work).