Obviously Cambridge is more prestigious than LSE as a whole but LSE is more prestigious when it comes to social sciences!
Gender studies would probably have more expertise at LSE.
So look at the overall aspects of the course structure and Location: ancient buildings vs a capital metropolis!
either way u can't really lose! depends on ur career goals too, e.g if ur trying to impress the average layman or employer than Cambridge will have that extra wow factor but if ur intent to work in social research, academia or impress an actual social scientist then LSE will edge it.
Either way there's no difference really!
I have notice that ur an international at that may be an additional factor as to why ur asking LSE or Cambridge form my experience LSE and Oxbridge have an equal international reputation but in the UK, as a whole, Oxbridge is almost untouchable, bar economic and politics at LSE.
Its a historical and cultural things that reminds us British of when Britain was the superpower in the world more than an actual reflections of the current reality. (those for example that still think Oxbridge can match Harvard Yale and Stanford)
So have this in mind to, back in ur own country LSE Cambridge will not be that different in the UK, except for social scientists, it will be.
i think u should factor these variables into ur decision! and also which one is cheaper