Betwixt is an archaic preposition meaning between. It sometimes appears in the cliché betwixt and between, meaning in an intermediate position or neither one thing nor another. Here and in all other uses, betwixt sounds archaic and is best avoided by writers who wish to sound modern and unpretentious.
Examples
In these examples, well-meaning writers attempt to use betwixt to enliven their prose:
In the area from the foul line to the no-charge arc, betwixt the daintiness of pull-up jumpers and the clatter and clash of driving layups, Jamison thrives. [Cleveland.com]
Betwixt the glamorous stores of Fifth Avenue, New York City, one store in particular is showcasing something other than glittering jewels and silky fabrics . . . [Daily Illini]
I’d pay good money to be photographed betwixt them! [Accidental Sexiness]
In each case, between could replace betwixt with no loss of meaning.

