Jack Sprat could eat no fat, his wife could eat no lean; and so betwixt the two of them, they licked the platter clean. Perhaps you've always said "and so between the two of them" when reciting the tale of Jack Sprat and his wife. That's fine. "Betwixt" and "between" have similar origins: they both come from a combination of "be-" and related Old English roots. Both words appeared before the 12th century, but use of "betwixt" dropped off considerably toward the end of the 1600s. It survived in the phrase "betwixt and between" ("neither one thing nor the other"), which took on a life of its own in the 18th century. Nowadays "betwixt" is uncommon, but it isn't archaic; it's simply used more consciously than "between."
First Known Use of betwixt
before the 12th century, in the meaning defined above
History and Etymology for betwixt
Middle English, from Old English betwux, from be- + -twux (akin to Goth tweihnai two each) — more at between