הלכה על סוכה 93:3

We have already said many times that the prohibition of work on all of the appointed times is the same. However I need to expand on the discussion here and inform you, my son, about the matter of this holiday which the Sages, may their memory be blessed, informed us (Yoma 2b), which is that it is a holiday on its own - meaning, that it is not part of the holiday of Sukkot. And even though you see all of Israel sitting in their sukkot on one of the two days of the holiday of this holiday of Shemini Atseret - it is not because it is a part of the holiday [of Sukkot]. As behold, we explicitly say in its blessings, "the eighth day of the holiday of convocation," and there is no mention of the holiday of Sukkot in it at all. But because of the ordinance of the two days of the exiles, we must sit in the sukkah eight days; and seven - like the law of the Torah - will not suffice. And therefore we sit in the sukkah on the eighth day of the holiday of Sukkot. And they, may their memory be blessed, said (Sukkah 47a), "We do sit" in the sukkah, in order to fulfill the obligation: Since they, may their memory be blessed, obligated us to add a day for each and every appointed time, we have also added to Sukkot and made the days of [the] sukkah eight days. But nonetheless, "we do not bless" over the sukkah on this day; as it it is truthfully a different appointed time - since we are now experts in the fixing of the [new] moon. And it is more fitting for us to bless for the holiday of the convocation which is true than for the other which is because of the ordinance. And perhaps you will say, "Why did they not ordain to recite a blessing for both of them - and we would say, 'on this holiday of Sukkot and holiday of the convocation' - as on Shabbat and an appointed time, we mention both of them?" Behold, this is not [correct]. As Shabbat and an appointed time [can] both of them be as one day, but it is impossible for two appointed times to be together. And hence it is not fitting for us to recite the blessing like this. But it is fitting for us to sit in the sukkah - as sitting in the sukkah does not detract at all from the holiday of convocation; and there is in the thing, [a fulfillment of] the commandment of the ordinance of the two days of holiday of the exiles which they, may their memory be blessed, ordained on every appointed time, as we have explained above (Sefer HaChinukh 301).