According to the Oxford, Webster, and Pharos dictionaries, both while and whilst have the same meaning. While is the older, and whilst has since largely fell out of favour. ‘Whilst’ is no longer used in the US, and – whilst still present – is not preferred in British English.
Personally, I find both have their uses. Now, I need to clarify that there is no official semantic distinction between the two, what I am about to tell you is just my personal preference and usage.
I prefer to use ‘while’ for the standard meaning, that, to indicate that two things are happening at the same time. ‘Whilst’, however, I use to indicate that, whilst things are happening at the same time, they are either unrelated, contradictory, opposite, or counter-intuitive, etc.