There is no difference in meaning between among and amongst – these two words can be used interchangeably.
However, amongst is less common in everyday use than among and is considered slightly more educated. In spoken American English among tends to be used almost exclusively; however in writing amongst can still be found.
The n-gram below shows the relative use of these two words over the past 50 years or so. As you can see, among is far, far more commonly used than amongst.
One of the earliest form of the word is amonge which from the mid 13th century. By the 16th century in south of England the -st was being used (which was sometimes used to turn nouns into adverbs). It was maybe because of the growing historical importance of the south in terms of politics and education that the -st version was seen as more educated.
Similar word pairs which also have connotations of education are while vs whilst, and amid vs amidst.