The eighth day of Succot, Shemini Atzeret, is its own festival. Outside Israel, it has to share the stage with the extra day of Succot added on for diaspora Jews to avoid doubt about the proper date of the holiday. This is the only time in the Jewish year when two festivals overlap. How do we observe both holidays without compromising the integrity of either one?
In Israel, we do not eat in the succah on Shemini Atzeret. Outside Israel, we eat at least something in the succah on Shemini Atzeret to mark the extra day of Succot.
(There are different customs about what and how much to eat in the succah on Shemini Atzeret, ranging from merely making kiddush at night to eating both Yomtov meals there.) But we do not make the blessing for sitting in the succah on Shemini Atzeret, as that would formally define the day as Succot and be an insult to Shemini Azteret. As a Chabad rabbi explained it to me by way of analogy, “how would you feel if a couple invited to your son’s barmitzvah decided to celebrate their wedding in the middle of your party?”