The only real difference between them is that 'whilst' is not used in American English and is, indeed, quite rare even in British English. Here, they are synonymous.
I just did a search on The Times' website... 202 whilsts, 16,412 whiles. Whilst just seems old-fashioned these days. But as QM says, they mean the same thing and there's no general rule... though I suspect there may be a rule in The Times style guide telling Times journalists which one to use.
Yes, J...the University of Lancaster's word-frequency list reveals that, in every million words of written English, 'while' appears 503 times on average and 'whilst' appears 58 times. In other words, for every one 'whilst' you're likely to see about ten 'whiles'. Not quite as different as The Times website's figures, but still justifying my earlier comment about 'whilst' being "quite rare".
well, as I said, QM, they may officially be an endangered species at Wapping, which would skew the figures. In that case, the list you quote would certainly be a fairer picture. (Incidentally, although a million words seems like a lot, I believe it's only 2 or 3 days' copies of The Times... perhaps a bit more since it shrank.)
Ah, H, but what about the old Monty Python line in the audio version of The Life of Brian which opened, "During the meanwhilst..."? I frequently use it to this day.
(This is a joke, by the way...not one to take seriously.)
Whilst should actually be written Whil'st because it's original form was Whilest. However, i'm confused while in middle english they had both Whiles and Whilest which seem to mean the same thing - thus must have been used in a different context, i just can't figure out what.