Both of these seem to describe very similar flows: curve-shortening flows of a high-dimensional hypersurface, which may form singularities, where the flow velocity at any point is normal to the surface and depends on the curvature there.
I guess it comes down to the difference between Ricci curvature and mean curvature. What are these differences? How does it affect the resulting flow? In what situations would we use Ricci vs. MCF?
Bonus question is, how do these compare to other similar flows?