If the modifier is able to smooth the edge of the donut without being
applied, why does it have to be applied in order to get the higher
density needed for the droopy icing?
The modifier creates new virtual vertices, procedurally, from the positions of the geometry entering into the modifier. But these generated vertices cannot be edited using traditional means-- they don't exist on the object, they only exist thanks to the modifier acting on that object, and so you can't really edit them. To create your icing droops, for them to be narrow enough, you need to edit the position of vertices, so you apply the modifier to turn these virtual vertices into real, editable vertices.
What is the difference between increasing the mesh density by
increasing the number of major and minor segments of the torus versus
using the Subdivision Surface Modifier?
In that specific case? The overall difference will be small. The subdivided torus will have smaller radii than the primitive torus (we talk about subdivision causing "volume loss," something common to a great number of 3D operations.) This volume loss will be different inside the torus from outside, distorting our cross-section from a perfect circle. If we start with a square torus, 4 segments, that torus will remain slightly square.
If we look at other shapes, there'll be a bigger difference. A subdivided cylinder or cube is a very different shape than you can get with any generation settings.
C-C subdivision just makes vertices in a different position, using a different algorithm, than primitive generation. A torus is a circle, extruded along a circle; its positions are determined by polar coordinates relative to the object origin or the center of that circle we're extruding. C-C subdivision doesn't care about object origins or polar coordinates; it generates positions on the basis of existing faces and their neighbors. You can read about the base algorithm at https://en.wikipedia.org/wiki/Catmull%E2%80%93Clark_subdivision_surface if you'd like.