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".
english.stackexchange.com/users/80/Matt Hamilton
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.)
english.stackexchange.com/users/3752/AAT
While and whilst annoy me.
Same goes for among and amongst.
As far as I'm concerned, the number one rule should always be don't opt for the -st equivalent simply because it sounds more distinguished.
Generally, I find 'whilst' more befitting when preceding a present participle.
With that said, the following take on your example seems a tad more agreeable: Whilst walking down the street, the sun was shining.
english.stackexchange.com/users/54/David Foster
It seems they are interchangeable, but whilst is primarily used by the British.
english.stackexchange.com/users/17/aphoria
"whilst" is "while" plus the suffix "-s" (with "-t" added by analogy with "amongst", "amidst").
The earliest citation in the OED for "whilst" as a conjunction meaning "while" is 1375.
So maybe that's the answer to your question.
english.stackexchange.com/users/13605/morphail
I have served as an editor for a scientific journal for many years.
In my observation, authors who use British English spellings also use "whilst" and "amongst".
The journal is published in North American English.
We consider these British spellings and always change them to "while" and "among".
english.stackexchange.com/users/278620/user278620
I know this is a dead thread, but it still appears in current searches, so I add this comment in the hope it comes back to life!
Although ‘while’ and ‘whilst’ can share a meaning when used as a conjunction, only ‘while’ can be used as a verb (‘while away the hours’) and a noun (‘this could take a while’).
Also, ‘whilst’ conveys the meaning of ‘alternatively’, or ‘but’, or ‘whereas’, however ‘while’ means ‘at the same time’.
So ‘whilst’ follows a comma, and ‘while’ does not.
Example...
“I told jokes while she ate lunch”.
We could both be eating lunch together.
“I told jokes, whilst she ate lunch”.
It is now clear that I am not eating lunch, and she is not telling jokes.
Another example where only ‘whilst’ works to convey a comparison...
“I watch action movies, whilst you watch dramas”.
This means that generally I don’t watch dramas and you don’t watch action movies.
“I watch action movies while you watch dramas” means I go and watch action movies whenever you are watching dramas.
I may love dramas...
I just don’t like watching them with you.
english.stackexchange.com/users/278662/Jeff Tribe
If there was ever a distinction in meaning -- that "whilst" meant "whereas; although" while "while" meant "at the same time as" -- it seems to have been lost by the early 19th century, as can be seen by the usage of "whilst" in the works of great writers, in contexts where the meaning of "whereas" would not fit.
For example: Whilst wandering on in this slow manner, they were again surprised ... by the sight of Mr. Darcy approaching them, and at no great distance.
-- Jane Austen, Pride and Prejudice, ch.43 Whilst he was watching the patient thus, the younger lady glided softly past, and seating herself in a chair by the bedside, gathered Oliver's hair from his face. -- Charles Dickens, Oliver Twist, ch.30 A cab had driven up whilst the American had been talking.
-- Sir Arthur Conan Doyle, The Adventure of the Dancing Men
english.stackexchange.com/users/187832/Rosie F
I think most folk happily use either "while" or "whilst". I've a vague recollection that at one time "while" indicated the passing of time and "whilst" was essentially the same as "whereas" or "although".  So using while for time passing...   While I was walking down the street the sun was shining.   ... and whilst for whereas/although...   Whilst I was walking down the street I often prefer to hop.   Any views?
When did "while" and "whilst" become interchangeable?
english.stackexchange.com
<word-choice><etymology><differences><conjunctions><historical-change>
english.stackexchange.com/questions/6
english.stackexchange.com/users/15/Nick Pierpoint
There is no clear agreement as to when "while" and "whilst" became interchangeable, but it is possible that "while" was the original term. If they did ever have different meanings, they probably lost them before the 19th century, and it is also possible that they have always meant the same thing, and it seems that "whilst" is primarily used by the British. They are more or less interchangeable, except that the -st version is more UK spelling and also only 'while' can be used as a verb.
There is no clear agreement as to when "while" and "whilst" became interchangeable, but it is possible that "while" was the original term. If they did ever have different meanings, they probably lost them before the 19th century, and it is also possible that they have always meant the same thing, and it seems that "whilst" is primarily used by the British. The earliest citation in the OED for "whilst" as a conjunction meaning "while" is 1375 and some argue that 'while' is the original version.  It seems they are interchangeable, but  whilst  is primarily used by in British English. Although ‘while’ and ‘whilst’ can share a meaning when used as a conjunction, only ‘while’ can be used as a verb and a noun. Also, ‘whilst’ conveys the meaning of ‘alternatively’, or ‘but’, or ‘whereas’, however ‘while’ means ‘at the same time’. Therefore, ‘whilst’ follows a comma, and ‘while’ does not.
While and whilst do mean the same thing and are entirely interchangeable. There is some disagreement as to whether or not the two terms have always been interchangeable. However, the earliest evidence of them being used interchangeably dates from 1375 and any difference in usage seems to have disappeared at the latest by the 19th century.
They are interchangeble but 'while' is the older version. Don't use the 'st' version just because it sounds fancier. They are interchangeable, but  whilst  is primarily used by the British. They are the same - "whilst" is "while" plus the suffix "-s" (with "-t" added by analogy with "amongst", "amidst"). whilst' is just the UK spelling. The difference is that 'while' can be used as a verb but 'whilst' cannot. If there was ever a distinction in meaning it seems to have been lost by the early 19th century.
A tip is that "whilst" is preferred if you think that starting your sentence with "while" could change the meaning. The earliest citation in the OED for "whilst" as a conjunction meaning "while" is 1375 and some argue that 'while' is the original version.  It seems they are interchangeable, but  whilst  is primarily used by in British English. Although ‘while’ and ‘whilst’ can share a meaning when used as a conjunction, only ‘while’ can be used as a verb and a noun. Also, ‘whilst’ conveys the meaning of ‘alternatively’, or ‘but’, or ‘whereas’, however ‘while’ means ‘at the same time’. Therefore, ‘whilst’ follows a comma, and ‘while’ does not.
There is some discrepancy as to whether they have always been interchangeable or if there was once a difference in use. The earliest evidence that they might have been interchangeable dates from 1375. If there was a difference in meaning, it seems to have been lost by the 19th century.