On a tool like a Delta 31-300, what is the advantage of having a disk sanding surface as well as a belt sanding surface? When would you decide to use one or the other?
An advantage of the disk over the belt is that you can control how aggressive the sanding is by moving the point of contact between the edge of the disk (where the speed is highest) and the middle (where it is lowest). The downside of the disk sander (pun intended) is you can only safely use the half of the disk that is pushing the work down onto the table. This means the disk works well for fairing convex curves. (If you need to sand concave shapes, you need yet another machine, a spindle sander.)
For smoothing long edges, the belt sander is better since you can work the full width of the belt. It would depend on the machine, but generally belts should be easier to change than adhesive disks.
Finally, by having the dual purpose machine you can set up both a coarse grit on the disk sander, and a finer grit on the belt sander to avoid having to change belts on a belt-only sander.