current york psych offer holder here - so i am gonna be biased
york is currently on top in the psychology rankings this year at 9th, but durham definitely isn't far behind as it's at 11th. also from what i can tell, people widely regard durham as an amazing uni but refrain from the same praise for york (if that matters to you). durham also dominates in the overall league tables at 7th whereas york is at 17th.
not too sure about the course at durham, but in york they offer to give you either a bsc in psychology (general) or a msci in a specific area - clinical, neuroscience or developmental disorders. york also offers the only way to get straight into a clinical psychology doctorate apprenticeship from taking your bachelors at an apprenticeship in nearly university of hull, so if clinical psych is what you want to do i 100% recommend york.
obviously there's a lot more to uni than the courses. look at the accommodations and cities and see which one fits closer to what you personally want, look at distance from home if that matters to you etc. york is a decently big city and the campus is as big as my hometown.
also: look at financing. this is a really big issue of mine for york personally since i dont qualify for anything but there seem to be quite a fair few scholarships and bursaries for many different people in durham which you may be eligible for - one of them is worth £13000!!!