Shemini Atzeret falls on the twenty second of Tishrei, the date this rule records, immediately following the seven days of Sukkot, and its name, the eighth day of assembly, already states the point rabbinic tradition insists on, that despite its position directly after Sukkot's seven days, it is treated as a wholly separate festival rather than simply an extension of the one before it. The distinction rests on the biblical text itself, in the book of Leviticus, which lists Sukkot's seven days and then a separate eighth day assembly with its own instructions, a textual basis the tradition reads as establishing two festivals rather than one prolonged observance.
The day's relationship to Simchat Torah, the celebration marking the completion and restart of the annual Torah reading cycle, differs by community in a way that mirrors the calendar's older Diaspora and Land of Israel split. In Israel, Shemini Atzeret and Simchat Torah are kept as the same single day, one observance carrying both the biblical eighth day and the Torah cycle celebration together. In much of the traditional Diaspora, where an additional festival day was historically kept as a hedge against uncertainty about the new moon's timing, the two commemorations occupy separate consecutive days instead, so a distinction the Bible draws between one festival and the next has, through calendar history, become a distinction some communities keep on one day and others spread across two.