The Torah portion for this week, Shemini, is from Leviticus 9:1–11:47 and the Haftorah from 2 Samuel 6:1–19.
The Sages wonder why the eighth day wasn't connected to the first seven, and they conclude that there is something unique about the eighth day of the inauguration. The rabbis wanted to stress the qualitative difference between the seven days and the eighth one, so they completely separated the readings.
For seven days, the representatives of Israel served in the Tabernacle, but on the eighth day, the priests began their service. This is the practical difference between the days, but there is a spiritual differentiation as well.