Both Stories and SPINs are packages to hold and deliver videos to the headset, but they serve different purposes.
A Story is the project of an interactive VR experience. A Story is created in SPIN Studio, where a user adds the sequence of videos or still images they would like to create a non-linear interactive story from. The Story is then paired to an Oculus Rift headset. In here the user can place graphics and hyper ports, change end-scene-behaviors, and set gaze interactions. Once completed, a Story is placed in a SPIN and shared out to be viewed by others.
A SPIN is the final delivery method of videos. It is essentially a playlist of videos, images, and/or Stories. A SPIN is made in SPIN Studio where the user can curate the selection of media. A SPIN's gallery experience can have its logo and home environment customized, and have auto play settings toggled on and off. A SPIN is also used to package guided experiences.