Amongst is a variant of among. There is no difference between them. While amongst is fairly common—though still rare compared to among—in British, Australian, and Canadian English, it is rare in American English and may even have an archaic ring.
The -st at the end of amongst is a holdover from a period of English in which s sounds were added to words (usually nouns) to make adverbs. Other examples of words inflected this way include always, once, whence, and unawares, and there are a few other -st adverbs such as whilst and amidst.
Though relatively rare, amongst is not absent from American English—for example:
Despite a year with no movie that distinctly stands out amongst its peers, David Fincher and “The Social Network” have been cleaning up at the early award shows. [The Massachusetts Daily Collegian]
While both the Wildcats and the Blue Devils are amongst the top five teams in the state, what happens from here on out is what really matters to both coaches and their wrestlers. [Charlotte Observer]
Amongst is unquestioned in British English, where it appears about once for every 18 instances of among. In these examples, note how the meaning of the sentence would be unchanged if among were used instead:
The overwhelming view amongst Scotland’s chattering classes is that Tommy Sheridan got off comparatively lightly in being jailed for three years at the High Court in Glasgow yesterday. [The Telegraph]
In and amongst his various reshuffles, Guidolin also had an epiphany. [Guardian]