There is going to be no major difference between these universities in terms of reputation. Durham might be slightly better but really t wont be a massive difference.
They are however very different places.
Bristol is a city whereas Durham is more of a town. It is only a fraction of the size of Bristol and things like nightlife or shopping will be better in Bristol than Durham. Durham has some of the best sports facilities of any university.
Durham is so small you can walk anywhere. Dependent on where you are staying you might need to catch buses in Bristol. Also Bristol as a city is very expensive and there are accommodation issues there. Bristol has been placing students in Newport and Bath due to a lack of accommodation.
Durham uses a college system whist Bristol does not. Durham will be cheaper to stay than Bristol and being in the North will be a degree or 2 colder.
Bristol is much more multi cultural than Durham although both will have many students from around the world.