There are more students from ARU than Brookes on TSR.
If you go back 30 years or more lots of upper and upper middle class girls used to do secretarial courses in Oxford. Although some did the same in Cambridge, Cambridge is a smaller town and was then less accessible to London. These were girls who would have a job until they married but who would never have a career and who would primarily be supported by mother and father until they married.
When secretarial courses went out of fashion in favour of degrees in the 1980s, this social set (including young men who were not academically strong enough to get into Oxford University but wanted to hang around their independent school chums and girls) started going to Oxford Polytechnic. Oxford Polytechnic was by far the most socially exclusive polytechnic. As it was fashionable, the grades required for entry were higher than for most Polys, which in turn made academic standards higher.
Cambridge didn't have a polytechnic until 1991; it had a College of Arts and Technology which was one step lower than a Polytechnic. It never attracted an elite social set. It simply provided vocational courses for locals. Indeed from 1989 it merged with a similar institution in the Chelmsford and was renamed Anglia Higher Education College.
In 1992 OxPoly became Oxford Brookes University and Anglia HE College became Anglia Polytechnic University and later Anglia Ruskin University.
Brookes had a head-start on ARU because it had the earlier and better Polyechnic. ARU has never quite made up the ground. Oxford is more of a city than Cambridge. There is little in Cambridge outside Cambridge University. ARU is still split between Cambridge, Chelmsford and now Peterborough. ARU has more vocational courses whilst Brookes has more general arts degrees, which is a legacy from OxPoly's popular courses in the 1980s. Brookes inherited a better campus than ARU. Although the elevated social class of Brookes students is far less noticeable than 20 years ago, it is still there to a slight extent. Brookes has 75% of students from state schools; ARU has 98%.
Generally speaking whilst Oxford and Cambridge are comparable universities, Brookes and ARU are not. For a degree in an arts subject, Brookes is significantly ahead of ARU. Having said that, both have subjects they specialise in, which either the other doesn't do or isn't known for.