They're both excellent, top 10 universities, and to be honest you'll struggle to choose between them based on teaching quality. According to the Times, Bristol betters Durham in terms of the student: staff ratio, amount spent on services/facilities and has better graduate prospects (both overall and for your subject), but Durham is ranked two places higher for Law and has better research. According to the Guardian, Bristol has a higher % of people satisfied with teaching score and again has better graduate prospects, but Durham is ranked slightly better for Law.
Have you looked at the courses and modules offered at both? Because even though they're more or less equal in terms of teaching quality, you may well find one course interests you more than the other.