As mentioned by banderas, 'whilst' has a more formal and literary quality. As such, even in Britain (perhaps the posh South-East excepted;-)) you won't here this substituted for 'while' in the sense of 'during'. So 'driving a car' and 'whilst' doing so 'eating a hamburger' is too mundane a situation - all that runny tomato sauce - for even a Britisher to come at.
However, with 'while' meaning 'despite the fact that'; 'although'
then 'whilst' might still be more commonly heard. Who wants to say the mouthful, "Despite the fact that he has many faults' when you can have, "Whilst he has many faults,...." -
'although' is such an overused word, it's pleasant to be able to vary one's conversation and throw in a 'whilst' every now and then.
Whilst I accept that he's not perfect in many respects, I do actually quite like the man.
Whilst I appreciate your point of view, I do also have some sympathy with...
Good conversation in good company and most acceptable!