In terms of Course, University ranking and Degree awarded etc. I'd place Durham and Edinburgh in the same category: one below Oxbridge, i.e. academically, both are excellent.
For me, this comes down to whether your DS would prefer to live in the more cosmopolitan city of Edinburgh, which will have a wider, more eclectic range of museums, theatres, cinemas, nightclubs, restaurants etc. than Durham ?
Durham is definitely smaller and has far less options in these areas - although it still has plenty (and probably enough for a student budget) and as a PP has mentioned, Newcastle is just up the road (10-15 minutes by train). Durham does have great pubs and (cheap) college bars though, if this is more his thing.
As PPs have also mentioned, the Durham collegiate system is unique. I went to Durham 30+ years ago and the friends I made in the my first year in college are still the best friends I have today.
Sport, Balls, Formals, Plays, Music and so much more, are all catered for in Durham colleges - they generate a real sense of belonging and camaraderie that friends who went to other universities just didn't get.
The private school/public school thing - for both places - is irrelevant IMO. He'll make friends with people you like and he';; find some will be from state, some from private and some from public.
Both are excellent choices to have - good luck with his decision.