I think most folk happily use either "while" or "whilst". I've a vague recollection that at one time "while" indicated the passing of time and "whilst" was essentially the same as "whereas" or "although".
So using while for time passing...
While I was walking down the street the sun was shining.
... and whilst for whereas/although...
Whilst I was walking down the street I often prefer to hop.
Any views?