Edinburgh is ranked significantly higher in the world rankings (17th in QS), does better on research REF rankings as you mentioned, does better in the UK league tables, has a higher entry tariff, has a much larger and much more famous alumni body, much higher endowment, is significantly older, has a much larger reputation abroad based on the university's prestige as well as the city itself, it is ranked 18th in the world by employers according to THE, much higher than Notts.
http://www.timeshighereducation.co.uk/news/global-employability-university-ranking-2013/2008497.article (2013 Edinburgh was 15th) You said that Edinburgh is a larger uni but actually they are a similar size, Notts 22k UG 13k PG, Edin 21k UG 11k PG
In nearly every subject Edinburgh is a better uni than Nottingham.
The only reason Edin doesn't produce as many law grads in MC is because its a Scottish Law school, it doesn't teach English law (you would need essentially a 4 year course + 1 year conversion vs 3 years at a English law school) and as a result isn't a very attractive option for those wanting to do law in London where MC is based. On the flip side its a top 2 law school in Scotland. It does well where its based in law and produces most of Scotland's top judges and the 3rd highest number of judges on the supreme court out of any UK uni.
http://www.scottishlegal.com/2015/04/24/law-graduates-from-uks-top-universities-can-expect-six-figures-salaries-after-ten-years/Look at these stats, after 10 years Edinburgh Law was ranked 3rd in grad earnings after Cambridge, LSE and ahead of Oxford.
The only thing Notts has over Edin is the fact that it has more recruiters on campus in the UK. That is it really. Worldwide employer reputation rankings all show Edin to be more recognizable than Notts. Nearly every ranking shows Edin > Notts and on all the hard stats including traditional factors of prestige like alumni, age, research income, endowment, nobel prizes, supreme court justices, prime ministers, scientists Edinburgh wins. Edinburgh does produce more CEOs if you look at the alumni lists, but if you have evidence that Nottingham produces more I am willing to change my opinion on that issue.