Globes are nice for teaching people the relation between a map and a spherical planet. They’re not necessarily the best at quickly answering geography questions, because you need to spin them around to make comparisons, and only the middle part of the globe (from any particular perspective) has low distortion. For e.g. data displays, they’re often quite misleading.

It’s difficult to follow rhumb lines on a globe, and it can also be difficult to follow great circle arcs (depending on how the globe is allowed to spin). Making arbitrary 3-dimensional rotations works pretty well for an unattached ball, but is very annoying to do using existing mouse + multitouch UIs. Comparing shapes, distances, and coordinates of places far apart on the globe is a pain. Globes (and computer displays) are difficult to blow up to very large size, and hard to annotate.

I think spheres are great (to the point that I made myself a spherical chalkboard), but having a sphere (or an interactive orthographic projection) doesn’t replace most uses of other maps.