In the real world, departmental reputation is probably far less important than it is made out to be here. Out of those two, for example, I'd go to Durham without a doubt, and in computer science I'm sure it'll be seen as stronger because it's Durham.
I'd say overall is more important, but if a department is crap then I'd steer away. But if a department's good but the university isn't, it wouldn't make me want to go there.
To apply it to my application, Durham is a very good university but rubbish for Business, therefore I chose not to apply there.