When did "while" and "whilst" become interchangeable?
Solution 1:
Always taking a bit of a chance using the Internet to answer an Internet question, but Daily Writing Tips says that not only are while and whilst interchangeable, but that in fact while is the original version. The very authoritative Michael Quinion backs this up on World Wide Words (and that is a site well worth visiting for anyone interested in the English language).
So I think your question proceeds from a false premise: they haven't come to be used to mean the same thing, they do mean the same thing, and there is nothing wrong with using while in all cases. (And to answer the question that wasn't asked, since it is shorter than whilst and clearly understood by US as well as British English-speakers, I'd say stick with while every time.)
Solution 2:
I read once that "whilst" is preferred if you think that starting your sentence with "while" could change the meaning. Consider this sentence:
While I walk I don't often whistle.
That could mean, "I walk, but I don't often whistle", or it could mean "I don't often whistle while I walk". Changing it to:
Whilst I walk I don't often whistle.
Removes that ambiguity.
I admit that the sentence itself is poorly constructed, but it does show one reason you'd use "whilst" over "while".