There is no difference at all - in any context.
Some people may tell you that 'everyone' is more formal or used more in written language, but this isn't normally a consideration for native speakers. We are generally unaware of which one we are using and tend to them use interchangeably, depending on what sounds best in the particular sentence. For example, if we're saying a sentence which has a couple of references to the word 'one', we might instinctively choose to say 'everybody' because it sounds better.
The same goes for anyone/anybody, someone/somebody and no one/nobody.