Hi Yamaha,
There's a spelling error in your question, so it should be "while" vs "whilst." There is one time when you can use either, but to be safe, you should use "while."
As a time adverb, you should use "while." For example: "You called while I was playing football." If you want to compare two things, you can use either "while" or "whilst." From what I've seen, it's quite a formal (and, in my opinion, pretentious) way of comparing things: "The chanterelle is a commonly found wild mushroom, whilst/while the champignon is commonly grown for commercial purposes."
"Whilst" seems to be an older version of the word—going back to Anglo-Saxon or at least Middle English roots—but that doesn't necessarily make it better.
I'd stick with "while," while keeping "whilst" in mind. Good writing is all about making sure your reader knows what you're talking about.