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.)