A double point is a stationary hammock and rigged via two points. It does not spin, and is perfect for yoga or conditioning resulting in the Aerialist to end up in the same position every time.
A swivel point comes with extra hardware and is rigged via a single point. However, it can also be used as a double point. A swivel hammock is most commonly used for aerial dance, allowing the aerialist to spin and show off position of the moves being practiced for all viewers to see.