The Sages tell us that the seventy bulls offered on Sukkot correspond to the seventy nations of the world, while the single bull offered on Shemini Atzeret corresponds to the singular nation of Israel. The Gemara illustrates: “This can be compared to a king of flesh and blood who told his servants, ‘Make me a big feast.’ Then on the last day of the celebration he said to his favorite, ‘Make me a small banquet, so that I can enjoy your company alone’” (Sukka 55b). A midrash makes a similar analogy. “This can be compared to a king who threw a week-long party and invited everyone in the country to it. Once the week was over, he said to his favorite, ‘We have already fulfilled our duty to the countrymen. Let the two of us continue with whatever we find – some meat, fish, or vegetable.’ Similarly, when God said to Israel, ‘On the eighth day you shall hold a joyous gathering for yourselves’ (Bamidbar 29:35), He was telling them, ‘Continue with whatever you find – a bull and a ram’” (Bamidbar Rabba 21:24).
Zohar (III 104b) presents a third variation on the analogy. All throughout Sukkot, Israel offered sacrifices on behalf of the seventy nations. Shemini Atzeret, though, is the King’s day to celebrate exclusively with Israel. It can be compared to a king who invited guests. Everyone in the palace worked on their behalf. Later on, the king said to them, “Thus far, you and I have made efforts on behalf of all the guests. You have brought all these offerings for the seventy nations. Now we will celebrate for one day, just us.” This is the meaning of: “On the eighth day you shall hold a joyous gathering for yourselves” – these offerings are for you.
Another midrash focuses on the joy of celebrating Shemini Atzeret with God. “When Israel heard about it, they began praising God with the verse, ‘This is the day that the Lord has made – let us delight and rejoice in it (bo)’ (Tehilim 118:24). Rabbi Avin asked: [The Hebrew word bo is ambiguous, so] we do not know whether to rejoice on the day or to rejoice in God. King Shlomo came and resolved it: ‘Let us delight and rejoice in You’ (Shir Ha-shirim 1:4) – in You, through Your Torah; in You through Your salvation…” (Yalkut Shimoni, Pinḥas §782).
This is what is unique about Shemini Atzeret. It has no special mitzva other than rejoicing in God, His Torah, and His salvation.