The Yom Tov of Shemini Atzeret is both a continuation of Sukkot and an independent festival. The fact that the Torah calls it the “eighth” (shemini) indicates that it is a continuation of the seven days of Sukkot. Likewise, with respect to the mitzva to make a pilgrimage and offer an olat re’iya (pilgrimage burnt offering) and shalmei ḥagiga (festival peace-offering), it was deemed a continuation of Sukkot; one who ascended to the Temple and offered the requisite sacrifices on Sukkot did not need to offer them again on Shemini Atzeret, while one who did not offer the sacrifices on Sukkot could offer them on Shemini Atzeret (Rosh Ha-shana 4b).
On the other hand, in several respects, Shemini Atzeret is considered an independent festival. First, the special mitzvot of Sukkot do not pertain to it: There is no mitzva to sit in the sukka, to take the lulav and etrog, or to offer a water libation with the tamid offering. Therefore, it has a different name; it is not called Sukkot, but Shemini Atzeret in the prayers, kiddush, and Birkat Ha-mazon. Second, the sacrifices offered on Shemini Atzeret in Temple times were different. On each day of Sukkot, fourteen lambs and two rams were offered, but on Shemini Atzeret, seven lambs and one ram were offered. On Sukkot, 13 bulls were offered on the first day, 12 on the second, and so forth until 7 bulls were offered on the seventh day. If Shemini Atzeret were a continuation of Sukkot, presumably 6 bulls would have been offered. In fact, only a single bull was offered, indicating that Shemini Atzeret is an independent festival (Bamidbar 29:32-39).
Since, in some ways, Shemini Atzeret is a holiday in its own right, we recite the berakha of She-heḥeyanu in kiddush at night; the She-heḥeyanu recited on the first night of Sukkot does not cover Shemini Atzeret (Sukka 47b; SA 668:1).