The first part of the holiday is marked by our dwelling in the Sukkah. This part of the holiday is shared with the seventy nations of the world, as we offer seventy cows (representing the seventy nations) over the course of this seven day period.
The second part of the holiday does not delineate where we need to live, whether in our homes or in sukkoth or even outside. This second part is special for Bnei Yisrael, a separate quality-time that the other nations don't celebrate.
What does this two-part holiday come to teach us? In the first part, we throw out one system in favor of another system. We move out of what we consider to be the constant world, and into another system called the temporary world, but it's still a system.
What do I mean? If you want to keep the mitzwah of Sukkoth, you have to live in a Sukkah. We exchange the system of permanent home with the system of Sukkah. If the goal was simply to throw out the permanent system, why don't we live outside, exposed to the elements? Why do we live in a temporary structure?
The fact that the first part of this holiday is tied to a systemic mindset is why the seventy nations get to take part in it. The nations, as we know, each have a prince, a guardian angel, through whom all their divine blessing flows.
Unlike the seventy nations, since Avraham Avinu, we Jews can opt out of the system and receive our divine blessings straight from HaShem, the Holy One Blessed Be He, Master of the World. This without any middle men.
This is why the second part, the final eighth day of the holiday is special only for us. On the eighth day we throw out systems altogether and connect with HaShem directly.
This holiday teaches every Jew that there is nothing standing between you and HaShem.