One of the major differences is with regard to the festival offerings. During the seven days of Sukkot, 70 bullocks altogether were sacrificed in the Temple. On Sh’mini Atzeret there was only one.
The Midrash explains: on Sukkot, Israel offers sacrifices on behalf of the nations of the world, which in those days were thought to be 70 in number.
By the end of the festival, we see that the nations are uninterested in whether anyone is praying and sacrificing for them, so God tells Israel to concentrate on itself and offer one bullock in its own name.
It is like a king who holds a series of banquets for his courtiers and citizens. After the series he says to his most intimate friend, “Come, let us have a banquet for just the two of us. We have been through so much together and done so much for others, but they still seem apathetic. It is time to celebrate between us, regardless of outsiders.”
The Jewish people try hard to be a light unto the nations, but the nations prefer their semi-darkness and there are nations and leaders that only want to wipe Jews and Israel off the map.
God’s tragedy is that the nations are happier in a comfort zone without too much of the Divine precepts of truth, justice and peace.
Not that Israel and the Jewish people are perfect. But they are perfectible and do their best to honour the God-given covenant.
For the others, God is often not even part of the agenda.