"Whilst" is a much more formal and literary way of saying "while". In daily life, I don't think I've ever heard anyone use "whilst" in place of "while". Even in writing, the usage is very rare.

I found this explanation helpful (ok, at least the first half). http://blog.oxforddictionaries.com/2016/02/while-or-whilst/

"Whilst" is a conjunction. "While" can be a noun, a verb, a preposition (extremely rare), an adverb, and a conjunction. I personally use both, and I know that both are used in the UK. While as a noun: A while ago, I bought a car. While as a verb: Reading is a good way to while away the time during the voyage. It means to spend time. As an adverb: The time while the animals eat. It means "during which". As a conjunction: I want a cake, while/whilst you want sweets. It means "whereas". Keep in mind that depending on the context, the meaning of while and whilst can slightly change; however, the meaning remains similar. Hope that helps.

There is no difference in meaning. Whilst ... I have the greatest respect and affection for my fellow Anglophone Jerry, can I just point out that the distinction "Whilst is British English.While is American English" is not strictly accurate? "Whilst" is not "British English". It's simply an alternative to "while" which happens to be much more common in British English than in American English. In the USA, you will generally only come across "whilst" in language ( books, plays and so on) which date from a much earlier era. In Britain, however, you may hear and read "whilst" being used in modern English. But .... and here's a fact which Jerry maybe doesn't know.... "while" is actually more common than "whilst", even in Britain. In British English, "whilst" tends to be used in formal written and spoken contexts. For example, many stations in the UK have a recorded announcement saying something like "Please keep your personal belongings with you at all times whilst on the station". In everyday conversation, some people use it, and some don't. I've noticed that I tend to say "while" in spoken English, but my children say "whilst". I have no idea why. It tends to be used by more educated people, but that isn't always the case. :)

Whilst is British English. While is American English. Interestingly, I was told by an Englishman that even they find 'whilst' to be old-fashioned and tend not to use it a lot. Maybe a real Englishman of Englishwoman will weigh in on this one. "Whilst" hasn't been used in American English since maybe 1910 or so.