There are literally a
billion threads already on this matter, but oh well.
The two universities are near indistinguishable in terms of grad prospects in the UK, be it in terms of the classic "employment within x months", presence in the City, or grad salaries after five years. UCL offers all the opportunities LSE does in terms of mooting, pro bono, career fairs and so on, so I can't imagine that choosing either will significantly impact you if you choose to work in the UK.
As someone who made this decision last year, and spoke to others who also found themselves in a similar position, I have to say that it usually boils down to the following preferences:
LSE > UCL: Greater international alumni network in high finance, law, etc (particularly HK and Singapore), a definite edge in reputation (varies depending on the country you're interested in), smaller university, more centrally located with more landmarks (RCJ, Strand, Temple) close by, higher concentration of City-oriented students, more numerous and specialised finance-related societies, slightly higher average salaries for both Law and non-Law (although beware that we're not comparing like for like here), most consistently nice-looking interiors across all buildings.
UCL > LSE: Campus feel, quieter residential neighbourhood with more students present overall (including those from SOAS and Birkbeck), more consistently sociable (?), students studying a wider variety of courses (e.g. those doing STEM or arts degrees), accommodation halls closer on average to campus, dedicated Law library (which I've happened to use once or twice, it's almost always full), newly-renovated Law faculty building to open in 2018/9 (Bentham House).
While I've addressed the question of unhappiness at LSE elsewhere, it ultimately boils down a few issues/groups of interest:
a) internationals who were promised the moon in terms of studying at LSE and realised that it didn't meet their expectations
b) all students doing Econ courses that are generally speaking not taught by god-tier academics, or who realise that the teaching in that department isn't great
c) people who make a huge fuss out of LSE100 and hate the fact that it wastes two hours of their week (1 for reading, 1 for the classes)
d) people doing humanities degrees that don't lead into a nice established career path (e.g. Sociology, Geography, History, Philosophy), who then go on to have a panic attack in their second/third years when they realise that most people around them already have job offers at top firms
e) people who made the entirely avoidable mistake of choosing to live at intercollegiate halls or High Holborn (both of which aren't very sociable)
f) people who become intimidated or insecure by the sheer prowess of their fellow students' CVs and feel that they can't match them in any way - that feeling of being unable to compete then leads on to people lambasting the culture for being "too competitive"