JS gets it mostly right on this topic. To add to it:
1/genuine atmosphere of debate and intellectual excitement;
2/constant, high level, involvement with the outside world;
3/hard-nosed, bracing approach;
4/cosmopolitan students and faculty;
1/the School's ambiguous position in the UK: there is a gulf between how LSE, and people across the globe, see the institution and how people in Britain view it. Its position resembles that of Imperial: Oxbridge is pissed off because LSE deflates the view that only Oxbridge matters, the vast majority of other universities are pissed off because they can handle coming second to Oxbridge, because Oxbridge somehow exists in another world, but they don't like the idea that this spikey, arrogant institution in Houghton Street can make such waves;LSE is the whipping boy among elite British universities;
2/the real estate: despite the expenditure of enough money to build several new campuses in greenfields suburbs the LSE 'campus' remains a dismal ragbag of buildings. Things are steadily improving, and the refurbished library is superb, but even yer average ex-poly would look down its nose at what is offered, in architectural terms, just off the Aldwych..the Houghtonophiles among us are secretly entranced by what they see as the post-Dickensian charm of the various alleyways and side turnings, but to everybody else it just looks a mess...an interesting mess... but a mess..
3/it's not friendly: it's no good pretending that people are always willing to pop round for a coffee and a chat about setting up a save - the - badger website....everybody's too damn busy..
5/the social sciences: the downside of this field of studies (which in LSE's case includes subjects like history, law, geography and so on) is there's no escape: it's real world time - whenever you open a newspaper there's something there to remind you of whatever it is you are supposed to be in the library studying...now if only LSE taught Aramaic...