My modern (and somewhat artificial) county sort of does. We have Newcastle and Whitley Bay to the north and Sunderland and Houghton le Spring to the south. There is some poverty in Newcastle (though not in the north of the city, mostly that's Gosforth which is well to do), and some affluent bits to Sunderland (but that has its own north-south divide; Seaburn, Roker, and Fulwell are the main nice bits to Sunderland and they're all in a little northern corner of town!).

Durham doesn't really though in the same way, it's basically city versus everything else. No matter which definition of the county you use (there are about three or four) you'll struggle to find one half better off than the other.

Northumberland though? Textbook divided county. The South East? Blyth, Bedlington, Ashington - that's all very deprived and ex mining or ex industrial. The rest of the county in the north and west is mostly small rural towns which tend to have less poverty. There's even a band separating the two - Amble, Morpeth, Cramlington - which is sort of intermediate status: these aren't necessarily well off but they're much more middle class and socially mixed than somewhere like Ashington.