Based on the teachings of the Nesivos Shalom, the Slonimer Rebbe (Simchas Torah Ma'amer 1)
(33) The LORD spoke to Moses, saying: (34) Say to the Israelite people: On the fifteenth day of this seventh month there shall be the Feast of Booths to the LORD, [to last] seven days. (35) The first day shall be a sacred occasion: you shall not work at your occupations; (36) seven days you shall bring offerings by fire to the LORD. On the eighth day you shall observe a sacred occasion and bring an offering by fire to the LORD; it is an Atzeres: you shall not work at your occupations.
(35) On the eighth day you shall hold an Atzeres; you shall not work at your occupations.
(6) Let me be a seal upon your heart, Like the seal upon your hand. For love is fierce as death, Passion is mighty as Sheol; Its darts are darts of fire, A blazing flame.
Rabbi Elazar said: These seventy bulls that are sacrificed as additional offerings over the course of the seven days of Sukkot, to what do they correspond? They correspond to the seventy nations of the world, and are brought to atone for their sins and to hasten world peace. Why is a single bull sacrificed on the Eighth Day of Atzeres? It corresponds to the singular nation, Israel. The Gemara cites a parable about a king of flesh and blood who said to his servants: Prepare me a great feast that will last for several days. When the feast concluded, on the last day, he said to his beloved servant: Prepare me a small feast so that I can derive pleasure from you alone.
(ד) דִּכְתִּיב (במדבר כט) בַּיּוֹם הַשְּׁמִינִי עֲצֶרֶת תִּהְיֶה לָכֶם, דְּהָא יוֹמָא דָּא, מִמַּלְכָּא הוּא בִּלְחוֹדוֹי, חֶדְוָותָא דִּילֵיהּ בְּהוּ בְּיִשְׂרָאֵל. מְתַל לְמַלְכָּא דְּזַמִּין אוּשְׁפִּיזִין, אִשְׁתְּדָּלוּ בְּהוּ כָּל בְּנֵי הֵיכָלֵיהּ, לְבָתַר אָמַר מַלְכָּא, עַל כֵּן אֲנָא וְאַתּוּן אִשְׁתָּדַּלְנָא כֻּלְּהוּ בְּאוּשְׁפִּיזִין, וּקְרַבְתּוּן קָרְבְּנִין עַל שְׁאָר עַמִּין בְּכָל יוֹמָא, מִכָּאן וּלְהָלְאָה, אֲנָא וְאַתּוּן נֵחֱדֵי יוֹמָא חַד, הֲדָא הוּא דִכְתִיב בַּיּוֹם הַשְּׁמִינִי עֲצֶרֶת תִּהְיֶה לָכֶם. לָכֶם: לְקָרְבָא קָרְבְּנִין עָלַיְיכוּ. אֲבָל אוּשְׁפִּיזֵי מְהֵימְנוּתָא, בְּמַלְכָּא מִשְׁתַּכְחֵי תְּדִירָא. וּבְיוֹמָא דְּחֶדְוָותָא דְּמַלְכָּא, כֻּלְּהוּ מִתְכַּנְּפֵי עִמֵּיהּ, וּמִשְׁתַּכְחָן. וְעַל דָּא כְּתִיב, עֲצֶרֶת, תַּרְגּוּמוֹ: כְּנִישׁוּ.
The verse states (Numbers 29:35) "On the eighth day you will have an Atzeres", for on this day Hashem is one with His people; He rejoices with Israel. It is comparable to a king who invites guests. The members of his chambers are busy taking care of the guests. Then the king says to these members, after you have spent all this time taking care of the guests and bringing offerings on their behalf, now let's have one day where only you and I rejoice. This is what is meant: "On the eighth day you will have an Atzeres", to bring sacrifices for yourselves. But the trusted hosts are always found with the king. And on the day of the King's rejoicing, everyone is gathered with Him. And that is what is meant by "Atzeres", which means "gather."
(ד) פָּתַח רַבִּי חִיָּיא וְאָמַר, (שיר השירים ח) שִׂימֵנִי כַחוֹתָם עַל לִבֶּךָ כַּחוֹתָם עַל זְרוֹעֶךָ וְגוֹ', שִׂימֵנִי כַחוֹתָם, בְּשַׁעֲתָא דְּאִתְדַּבְּקָא כְּנֶסֶת יִשְׂרָאֵל בְּבַעְלָהּ, אִיהִי אַמְרַת שִׂימֵנִי כַחוֹתָם, (ויחי רמ''ד ע''ב) אָרְחָא דְּחוֹתָם, כֵּיוָן דְּאִתְדַּבָּק בְּהַהוּא אֲתָר דְּאִתְדַּבָּק שָׁבִיק בֵּיהּ כָּל דִּיּוּקְנֵיהּ, אַף עַל גַּב דְּהַהוּא חוֹתָם אָזִיל הָכָא וְהָכָא, וְלָא קַיְּימָא תַּמָּן, וְהָא אִתְעֲבָר מִנֵּיהּ, כָּל דִּיּוּקְנֵיהּ שָׁבִיק תַּמָּן, וְתַמָּן קַיְּימָא. אוּף הָכִי אַמְרַת כְּנֶסֶת יִשְׂרָאֵל, כֵּיוָן דְּאִתְדַּבָּקְנָא בָּךְ, כָּל דִּיּוּקְנִי לִיהֲוֵי חָקִיק בָּךְ, דְּאַף עַל גַּב דְּאֵיזִיל הָכָא אוֹ הָכָא, תִּשְׁכַּח דִּיּוּקְנִי חָקִיק בָּךְ, וְתִדְכַּר לִי.
Rabbi Chiya opened with the verse in Song of Songs 8:6: "Let me be a seal upon your heart, like a seal upon your hand." At the time when the Congregation of Israel cleaves to its Master, she says "let me be like a seal". The way of a seal, once it sticks to a place, it leaves its mark, such that even if the seal is gone, the mark is still there. So too, the Congregation of Israel says, once we cleave to You, let our mark be left with You, so that even if leave Your Presence, our mark will still be engraved upon You, and You will remember me.
(7) Thus said the LORD of Hosts: If you walk in My paths and keep My charge, you in turn will rule My House and guard My courts, and I will place you, the one that goes, one among those that stand still.
Nesivos Shalom, Simchas Torah, Ma'amer 1:2
On both Yom Kippur and Shmini Atzeres we nullify ourselves to Hashem. On Yom Kippur, this is accomplished through prayers, fasting and teshuva, whereas on Shmini Atzeres, this is accomplished through joy and dancing.