Hi, I lived in London until I was 10 and then moved up to Manchester so hopefully I can be useful.
Manchester is a lot less busy on the whole. Less of a buzz I'd say and much more liveable with respect to traffic, prices, crowds. The people on the whole are a lot nicer - I take the train between the two cities pretty often and whenever someone strikes up a friendly conversation or even just looks at you and smiles, it tends to be a Northerner rather than a Southerner. I realise I miss the buzz of London and the joys of having so many areas to explore, culture from around the world, and opportunities and events when I visit London now but living in Manchester is totally fine. It's a big enough city but also much much more livable and it's nice to be able to go from bustling city centre to quiet leafy suburbs or out-of-town open spaces in only a few minutes. There's plenty going on around the city and Greater Manchester as a whole but you don't get spoiled like a Londoner does. Bare in mind also that many other northern cities aren't that far away on the train.
Public transport is definitely leagues below London. Ticketing is a pain because several bus companies operate their own tickets and passes and single fares are ludicrously expensive for adults. I pay around £2.40 for a single 20 min journey to Piccadilly from where I live but I guess these prices are what shifts their day passes. However, there are free "Metroshuttle" bus going around the inner city and fares for buses just going up and down Oxford Road aren't as high. As a student, most of your travel will be on "Magic Bus" Stagecoach services so you can just get a ticket for that but then if you want to go out into Salford, you need a First ticket, or into South Manchester, you'll need a Stagecoach pass for all of their services and so on. The trams are good, usually reliable and engineering works don't happen nearly as often as in London. The fares are lower than Oyster for tubes on the whole but you probably won't be using them regularly as a student due to where the tracks run and where campuses are.