Full disclosure before we get started; I did my undergraduate at Leeds and now I'm staff at Durham.
I can only really explain the differences between the two universities. I don't want to sway you either way because you'd need need to be happy with that choice.
From my point of view there doesn't appear to be a massive difference in the standard of teaching, and that's pretty much where the similarities end.
Durham is a much more traditional university and has an excellent pastoral support system through the colleges which results in it having an exceptionally low drop-out rate. Durham also allocates 'academic advisors' that are based in departments and, in order to separate 'home' and 'work' problems, they'd practically only deal with academic issues. At Leeds the pastoral support usually falls to personal tutors based in departments, and they'd signpost students to support services. Both Durham and Leeds have central support services (e.g. counselling), but at Durham the college staff are more willing to assist with smaller issues than personal tutors at Leeds usually would be.
Because Leeds is a larger university and in a larger city there are many more opportunities available to expand your horizons. Leeds has nearly 300 student societies covering everything from Anime to Zoology. At Durham you'd be able to find something to suit your interests; there are societies run in colleges and also through the Student's Union, but these might not be as wide-ranging as the offer at Leeds. If you're planning regular nights out Durham may get boring pretty quickly, but a lot of students travel up to Newcastle at least once. With Leeds all this is practically on your doorstep.
Undergraduate students at both Durham and Leeds usually stay in university accommodation in their first year. At Leeds this tends to be self-catered but there is a large refectory in the centre of campus open all day (8am - 9pm term time weekdays), so you needn't go hungry if you don't feel like cooking. At Durham the colleges tend to be catered with relatively fixed mealtimes (e.g. 2 hours for each meal service), although the colleges realise students would prefer this to be more flexible and that might be something they look into for the future.
In summary, Durham is a more supportive University than Leeds, but Leeds is a bit more dynamic and would allow you to expand your horizons more. That might not be too much of an issue with the course you're doing as I'd imagine you'll do a year abroad. The choice is yours!
Hope this helps :-)