Congrats on the scholarship! Durham is still better than Birmingham, and in the long run, I would think a Durham degree is still significantly better. If tuition would have been an issue for me I would have gone with Birmingham, but if my parents told me it is not a problem, I would have surely gone with Durham.
Ahh, of course, someone brought up Russell Group unis, one thing you need to understand is that Russell Group unis are not some sort of Gold standard, yes sure there is some top unis part of it but not all of them are the same level. Durham is surely better than Birmingham. Oxford/Cambridge/LSE is most definitely better than Birmingham.
In the business world, employers do differentiate between candidates based on universities, and Durham holds a significant edge over Birmingham in its business school programs. You have cherry picked one ranking table, Durham is a top 5 uni in the UK based on the majority of UK university rankings. Plus, league tables are mainly based on research output and student satisfaction.
I suggest the OP should go search online, type in Linkedin UK, and then the name of some select top companies and then either University of Birmingham or Durham University, and the difference will be clear. I would also suggest connecting with people in those companies and seeing what they think about which university you should choose.