Oh dear – I’m afraid this one makes me a bit sad! The pair was suggested by Gill (she of the Libro holiday) and I realised I wasn’t quite sure of the difference, so I turned to my reference books.
Gentle readers: there is no difference.
Whilst is a variant of while which is pretty well only used in the UK (it’s considered archaic, at best, in other regions). Some style guides, such as the Guardian newspaper one, say whilst shouldn’t be used at all. I like it, and I’ll continue to use it – but there are no rules about when you should and shouldn’t do so.
And it’s the same for among/amongst too. Sorry!
You can find more troublesome pairs here.