If there was ever a distinction in meaning -- that "whilst" meant "whereas; although" while "while" meant "at the same time as" -- it seems to have been lost by the early 19th century, as can be seen by the usage of "whilst" in the works of great writers, in contexts where the meaning of "whereas" would not fit. For example:
Whilst wandering on in this slow manner, they were again surprised ...
by the sight of Mr. Darcy approaching them, and at no great distance.
-- Jane Austen, Pride and Prejudice, ch.43
Whilst he was watching the
patient thus, the younger lady glided softly past, and seating
herself in a chair by the bedside, gathered Oliver's hair from
his face.
-- Charles Dickens, Oliver Twist, ch.30
A cab had driven up whilst the American had been talking.
-- Sir Arthur Conan Doyle, The Adventure of the Dancing Men