I lived in London for many years and I love it passionately as a city. (My Dad is a proper cockney and have lots of family who have lived there for centuries). I now live in Sheffield. I think I like it more up north, but it is not an unmixed picture.
The thing I love about living in the north is how easy it is. First of all, the difference in house prices means that I can afford to live extremely cheaply (I am a woman of very simple desires), which ultimately means that rather than being locked into a job that I can't leave for financial reasons, I can basically do whatever I like. The freedom it brings is amazing. I can never be grateful enough for it, really - to the point that it basically outweighs everything else/
Secondly, the city is so easy to get around. I love walking when I am well, and I can basically walk anywhere in Sheffield from my house. I can be in the city centre in 15 minutes, or in the peak district in 15. I can even walk to Hathersage across some of England's most beautiful scenery via footpaths that involve me crossing just a couple of roads. Sheffield is also the safest place I've ever lived - pubs are full of a very mixed crowd of all ages, and you don't get hassle as a woman. 
If high culture is your thing, the theatre is amazing. There's a fabulous independent cinema, too. But the galleries and museums are crap and you'll need to travel for classical music too.
However, there are things I don't like. It isn't as friendly as the south. People always tell me that Sheffield is a friendly city, but in my experience of living here for 10 years this is a total lie told by people who have never lived anywhere else and have no point of comparison really. It doesn't begin to compare to London let alone friendlier places in the South East. In London, you can walk into a shop and have a bit of banter with a shopkeeper you've never seen before and it's lovely. In Sheffield, people think you are mental if you speak to strangers in shops about anything except shopping. Some people in shops will sometimes growl at you or be rude like they don't really want to sell anything. There isn't much of a culture of smiling or chatting to strangers on the street either. I found this very depressing when I first moved here. There can be a culture of aggression too - not physical violence, but temper and frustration are probably shown a bit more than darn sarf. The upside of this is that there is no smarmy bullshit of the kind you sometimes get sold in London, and people don't tolerate pretentiousness.
Shopping in the city centre is rubbish - if you want big stores, you need to travel to Meadowhall, which in my opinion is hell on earth and to be avoided at almost any cost. However, there are peripheral areas with increasingly interesting and lovely independent stores, which are so much fun to shop in! Nottingham and Manchester are within easy reach, though, and fun to shop in if you need to visit the high street chains.
Food culture is improving slowly, but nowhere close to London yet. There are a handful of decent restaurants in the city centre, though, and a couple of stellar establishments close by. You can get a better pie than anywhere else in the country, though
The vibe of large parts of the richer areas of the city is conservative with a small 'c' (take a look at the interior decor of the richest areas in Sheffield on Rightmove and you see an explosion of Hyacinth Bouquetness). This is true architecturally too - the city feels like a lot of it was jerrybuilt, without the care and attention shown in other northern places, though to me this has a kind of charm now. Shops in the city centre shut at 5 and much of it falls rather silent - there isn't the sense of random adventure that you get in London, where you could meet just anyone stood outside a pub. On the other hand, the pubs are homely, safe, and populated by a proper cross-section of people, and the beer is fantastic - less so the wine or cocktails in most places.