Betwixt means the same as between. It is rare in the U.S., where it is considered an archaism, but it’s still used fairly often in British English, more often in speech than in writing. It sometimes appears in the redundant cliché betwixt and between, meaning in an intermediate position or neither one thing nor another.
Except where betwixt is still a living word, its use can come across as a sometimes cutesy, sometimes pretentious affectation. For instance, it is at least a little out of place in these American publications:
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]
[A]ll is not well betwixt the members of the Pineapple State delegation. [Vanity Fair]
In each case, between could replace betwixt with no loss of meaning.