Sh'mini Atzeres, although, as its name implies, is the eighth day since the beginning of Succos, and is consecutive to Succos, was generally considered by our Chachamim as a separate Yom-tov. The Torah (VaYikra 23:34) "...On the 15th day of this seventh month is the Festival of Succos, seven days for G-d."
Based on this the Gemara Succah (48a) calls Atseres a "holiday unto itself." In fact, the only connection made between Succos and Sh'mini Atseres in the literature is for the purpose of making up for neglected Chagiga offerings (Yoma 3a), which is also true for the day after Pesach too. For this reason, there is no obligation to take lulav and esrog, to sit in the Succah (except outside of Eretz Yisrael, where do to the uncertainty in the calendar, the two overlap), the Ya'aleh ViYavo and Kiddush have a different text, and only one cow was needed for the musaph offering (as opposed to the progression of 13, 12, 11... required for each day of Succos.)
In Shas, Sh'mini Ateres is called Atseres shel Chag (the Atseres of Succos), as opposed to Shavuos which is called Atseres without a qualifier (Minachos 65a a.e.). In fact, the Midrash (Shir HaShirim 7:2) takes the effort to explain why Sh'mini Atseres isn't 50 days after Succos, why it differs from Shavuos.
Similarly, Succos is compared to Pesach. As we just said, both play the role as the precursor to an Atseres. The fact that both are on the fifteenth of the month is no coincidence, numerous halachos are built upon that comparison (Succah 27a). Further, the mitzvah of Succah and the mitzvah of Matzah are compared (ibid). Just as matzah is obligatory on the first night of Pesach, and for the rest of Pesach it's only a restricted means of eating (one may eat baked goods only if they have not leavened), so too Succah, one is required to sit in it the first night, the rest of Succos one must only sit in the Succah when one wants to eat.
In fact, the parallelism starts even earlier in the season. Yom Kippur is called by the Torah "Yom HaKippurim", which is explained homiletically (Yoma 2a a.e.); in the days of the Moshiach it will become a "Yom", a day, "Ki", like, "Purim".
Thus, we find the Jewish year divided about two triads: one, Aseres Yimay Tishuvah which climaxes at Yom Kippur, followed by Succos and Sh'mini Atseres; the other, Purim, Pesach and Shavuos. The first before the winter, the second in the beginning of spring.
Succos is in celebration of divine sustenance. It is Z'man Simchaseinu, a time to be happy with our lot. It is before winter, when we gather in the grain that will allow us to survive the upcoming months. It has a universal theme - that of Hashem feeding and sustaining the world.
It can only come after Yom Kippur. We have just been judged, "who will live and who will die ... who will be in famine, and who in drought ..." We are confident our tefillos were excepted, and thus, we come to the Beis HaMikdosh to celebrate our lot.
This progression culminates on Sh'mini Atseres. It is a continuation of Yom Kippur, as we pray for rain in musaph. Tehillim 27, "LiDavid", a prayer for aid in our repentance is added to the end prayer until Sh'mini Atseres. It is also a culmination of our celebration of Succos. We rejoice in the role our role as the upholders of the Torah in insuring the world's existance. The seventieth musaph cow, the one corresponding to the B'nei Yisrael, is brought. We end and begin the torah, showing our continuing dedication to our responsibilities in exchange for this aid. We never pause at the completion of the Torah, we must go on - "for they [the words of the Torah] are our lives, and the length of our days." (The separation of Sh'mini Atseres and Simchas Torah outside of Eretz Yisrael is due to a historical confusion about the calendar. Both should be the same day.)
The sequence from preparation, to celebration, to culmination is also found in the other triad at the other extreme of the year.
Many times the Jews were in dire peril and Hashem saved us. Why was Purim chosen as a holiday and not another similar occasion. It says in Gimara Megillah "'The Jews confirmed and took upon themselves' (Esther 9:27) the Torah." Whereas at Mount Sanai we accepted the Torah under threat of having the mountain dropped upon us (Rashi Sh'mos 19:17), on Purim we finally excepted the Torah willingly.
We are the chosen nation only because we chose Hashem first (Rambam, Igeres Teiman). After we celebrate the finalization of the acceptance of the Torah, we celebrate Hashem's relationship with us. He took us out of Egypt "to be for Him a treasured people and a holy nation." The entire calendar is based upon the demand that Pesach be in the spring. Celebration of our birth as a nation can only be at the time of regeneration of nature.
This culminates with Shavuos. Hashem presents us with the Torah. This is the "wedding" between G-d and "Israel". Hashem only took the B'nei Yisrael out of Egypt only to give them the Torah. Shavuos is the Atseres of Pesach, without Shavuos, Pesach would lack meaning.
Only in this structure of the Jewish year, can we properly observe Sh'mini Atseres. It is an Atseres, a culmination. We end the season dedicated to the sustenance of the world. As opposed to Shavuos, a re-creation of the giving of the Torah, of G-d and Israel, here we take pains to show that we are rejoicing in the continual nature of Torah and mitzvos. We are celebrating the Torah as a source of sustenance for us and the universe at large. For this reason, we celebrate the continuation of Torah study on the second day of Shmini Atzeres, Simchas Torah. It is only our continual study and observance of the Torah that perpetuates the universe's existence.© 1995 The AishDas Society