For me it's really hard to define, it's a sort of feeling rather than tangible for the most part, but I get it, it's really irritating when Londoners smugly refer to 'the buzz' of the city that smaller towns just don't have without being able to logically explain what it actually is! As someone who's lived in both I'll have a go... nb I am not saying all of this is or should be universally wonderful and liked, it's all a matter of taste but for me it's what I mean by 'the buzz'
There's a certain anonymity to London which you don't always get elsewhere, people come and go, you don't always know your neighbours well and you are frequently in busy/crowded places surronded by lots of strangers when going about your daily business, which can be quite exciting/fun for those of a certain mindset.
It's also (IMO) quite untrue that Londonders are unfriendly, yes people are often rushing around not hanging around looking for a chat on random street corners or on the tube, but I would actually say it's vastly easier to make friends and acquaintances in London than almost anywhere else in the UK, with a little effort and in the right context you can easily meet and strike up conversations with a hugely diverse range of people from all over the world as London is hugely cosmopolitan and has a big transitory population of working-age people. Plus most Londoners, even the natives 
are actually quite friendly when you get to know them!
There are lots of shops, cafes, bars, restaurants everywhere, even in the more suburban parts of London, you are rarely far from a 24 hour shop or somewhere to buy a poncey coffee 
and even if you don't avail yourself of these facilities that often, just seeing them around gives a 'buzzy' feel, and also the shops frequently change so even on your familiar local high street there's usually something new and different to look at.
London also encourages spontaneity, the enormous range of cultural, artistic, sporting, social events and opportunities happening on a literally daily basis, and if you live pretty much anywhere in zone 1-6 you decide you want to attend an exhibition or go to a concert or take up an esoteric hobby or eating an unusual new cuisine, find it, book it and be doing it within an hour - you may not actually avail yourself of this that often, IME most Londoners are like everyone else and may be full of grand plans for the weekend or whatever but when it comes to it prefer slobbing on the sofa with a pizza and Netflix - but there very definitely is something in the feeling of knowing you have all the options. Even living within a hour's train journey of a major city where logically speaking I had very nearly the same options and possibilities as when I lived in London I somehow didn't have that same feel, yes there's perfectly good theatres and galleries and so on but not the same volumne and accessibility as London and there's no real difference between jumping on a train or the tube but it feels more of a faff and something you have to plan actively for and so on (I said it was illogical and feelings based!)