There is no difference in meaning between fiber and fibre. Fiber is the preferred spelling in American English, and fibre is preferred in all the other main varieties of English.
Both spellings are many centuries old, and neither spelling was clearly prevalent on either side of the Atlantic until the second half of the 18th century. This was a period in which many British educators began to consider it proper for English words of French and Latin origin to take their more French and Latin forms rather than their more Anglicized forms. Fibre is the French spelling of the word from which the English word is derived, so it was promoted as the standard spelling despite its being unphonetic. The belief that French and Latin should hold sway over English never had much traction in the post-independence United States, and while Americans also favored the French spelling through the 19th century, the more phonetic fiber steadily gained ground through that century until becoming the preferred form around 1910.
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