It is interesting that you have chosen these two universities as both are particularly popular with students from private schools. They are almost equally highly regarded for Economics and History and you would not be doing yourself a disservice if you chose Exeter. Assuming you do get an offer from Durham, my advice would be to visit both cities if you possibly can and spend some time looking at the Exeter campuses and the Durham colleges. As well as considering any difference in course structure and assessment methods, it's worth pondering how it might feel to live in and around each university, with particular regard - in the case of Durham - to any preferred college you had in mind and its location in relation to the buildings where teaching takes place for your chosen subjects.
It struck a chord with me when you said, "... as I already go to a boarding school, I don't want uni/ the college aspects of Durham to feel like a boarding school again". That's exactly how I felt, many years ago, when I chose Warwick for my undergraduate studies! Years later, I did a short postgraduate course at Durham, based at Hatfield College. Yes, it had something of that boarding school feel about it but I actually loved the place! I have often called Durham the 'Harry Potter version of Oxford' and I think that just about sums it up.
So, give them both a good look, talk to some of the current students and then make up your mind. At the end of the day, these are two excellent universities and you will prosper best where the course suits you and where you most feel at home.