Whilst is the alternate form of "while" that can only be used mid-clause.
"I'll do the dishes whilst you do the laudnry"
as apposed to
"While you were out of town, i skeeted all over your furniture."
An intelligent word to use in the place of "while"
I was trying to take a look at that hot chick whilst driving. It didn't turnout so well.
Anachronistic form of "while", used to create an illusion of intelligence or literacy. More often than not makes the user sound like a pretentious ass.
I always listen to Mozart whilst studying.
Centuries ago, the word "while" developed into "whiles" and "whilst". "Whiles" was soon dropped from usage, but "whilst" persists to this day, even though it is considered archaic.
The only difference between their meanings, and it is a subtle one, is that "whilst" has a slightly greater emphasis on two events happening concurrently. For example:
While other goods are becoming cheaper, bread is becoming more expensive." (You would not use "whilst" here.)
The guys were filming the video whilst jumping off cliffs! (You could use "while" here, but you would lose the emphasis on the dramatic fact that the two acts were being done at the same time.)