The OpenSubdiv manual uses terms “subdivision” and “tesselation” to address these differences:

http://graphics.pixar.com/opensubdiv/docs/subdivision_surfaces.html#subdivision-versus-tessellation

The “subdivision” is the Catmull-Clark algorithm as it is. It is an iterative process. On each step, each quad of a control cage is subdivided into 4 and the resulting surface is smoothed out. And then the same process is repeated while using the result of the previous step as the control surface for the new one.

Each step brings the surface closer and closer to the “limit surface”, but it is never reached.

Key factors here:

- The “subdivision” is iterative. Applying the subdivision of level 2 is the same as applying the subdivision of level 1 twice.
- The positions of the vertexes get closer and closer to the limit surface, but they never reach it. Typically, the surface shrinks with each iteration. The smoother the surface, the smaler this shrinking effect. So the change in size is less for each further iteration.
- Each iteration (except first) multiplies the dencity of the subdivided surface by 4. Each quad turns into 4 after each level. For example, level 0 - 1 quad; level 1 - 4 quads; level 3 - 16 quads, level 4 - 64 quads, …

The “tessellation” is acheived by representing the limit surface of the Catmull-Clark algorithm as a set of spline surface patches (in trivial case, each quad converges to a B-spline patch). And positions of points on these patches can be computed directly, without iterative refinement of the whole cage.

Key factors here:

- The vertices of the tessellated surface are always on the limit surface. The positions of the corresponding vertices remain stable regardless of the level of tesselation. The surface does not shrink (well, actually, in a sence, it shrinks once, when going from the control cage to the tesselated surface)
- The tesselation is not iterative. Tesselation of level 2 is not the same as applying the tesselation of level 1 twice.
- The density of the tesselated surface does not have to be controlled by the tesselation level. The number quads resulting from a quad on a limit surface does not have to be a power of 4. Nothing prohibits from, say, tesselating each quad into 9 (each edge into 3) or 25 (each edge into 5).
- The density does not have to be uniform (different quads of the control cage can have different density on the tesselated surface).

Most of the 3d software (including Blender up to 2.7) uses “subdivision” in their “subdivision surface” modifiers.

The blender 2.8, for some reason, decided to use what essentially a form of “tesselation”.

In general, the “subdivision” and its iterative nature is useful during modeling. For example in workflows that generate a base shape using a coarse control cage, then apply a couple of iterations of the subdivision surface and then modify the resulting topology to model the finer details.

(The current Blender’s multires does not facilitate such workflows since many refinements require changes in topology)

The “tesselation” is useful when applied to a final control cage (when subdivision surface is used purely for controlling the level of detail of the resulting surface), since it produces more sable results (no shrinking). This includes all forms of adaptive subdivision, obviously.

In my opinion, the “subdivision” (pre 2.8 version of subdivision surface) must return to Blender.

There must be an option to disable snapping to limit surface.

It is completely ok if it comes at the cost of disabling the GPU acceleration for modifiers that have snapping to limit surface disabled. Acceleration is never needed in such cases anyway.