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.
For example: Whilst I do like to use ‘whilst’ in my academic writing, ‘while’ is far more popular with others.
In the above example ‘whilst’ is used to indicate that the two topics (my use of ‘whilst’, and the popularity of ‘while’ with others) is opposite one another.
This, I suppose, reflects somewhat on my general ethos when it comes to diction and word usage: if the words are there, why not use them? The more expansive and diverse your diction, the more nuance you can express, and that is very important in the world of academic writing where concepts and theories are riddled with unbelievable nuance and complexity. Rather than pick between ‘while’ and ‘whilst’, or ‘cannot’ and ‘can not’, and use the chosen one to cover all the meanings it could have, why not use both, and each for a particular nuance of the broader meaning?
A little bonus
When it comes to ‘cannot’ and ‘can not’ I use cannot to indicate the impossibility of the actions, and can not to indicate the possibility of either performing the action, or not performing the action. I use them this way because because (the repeat is not by accident) the ‘not’ is embedded in the word ‘cannot’ the negative is implicit to the term, i.e. no matter how you say it (stress, tone, speed, etc.) it still has the same meaning, its meaning cannot change. Can not, on the other hand, has the negator separate from the ‘can’, thus through the variation of tone, stress, and speed the meaning of can be changed.