This is an excellent writeup but I think saying that (multilevel) DD and MG have a lot in common is not accurate, or at least not useful. The methods are very different and I don't think that expertise in one is very useful in the other.
First, the two communities use different definitions of complexity: DD optimizes the condition number of the preconditioned systems and MG optimizes work/memory complexity. This is a big fundamental difference -- "optimality" has a totally different meaning in these two contexts. Things don't change when you add in parallel complexity (although you get a log term added in MG). The two communities are almost speaking different languages.
Second, MG has multilevel built into it and multilevel DD methods have all been developed with two level theory and implementations. This limits the space of coarse grid spaces that you can use in MG -- they must be recursive. For instance, You can not implement FETI in an MG framework. People do some multilevel DD methods as <PERSON> mentioned but at least some of the current popular DD methods do not seem to be implementable recursively.
Third, I see the algorithms themselves, as practiced, as very different. 57
Qualitatively speaking I would say that DD methods project onto domain boundaries and solve this interface problem. MG works directly with the native equations. Avoiding this projection allows MG to be applied to nonlinear and unsymmetrical problems easily. Although the theory all but goes away for nonlinear and unsymmetric problems they have worked for a lot of people. MG also explicitly decouples the problem into two parts: the coarse grid space for scaling and an iterative solver (the smoother) to solve the physics. This is critical in understanding and working with MG and is an attractive property to me.
Although theoretically the smoothers and coarse grid spaces are tightly coupled, in practice you can often swap different smoother in and out as an optimization parameter. As <PERSON> mentioned point or vertex smoothers are popular and usually faster, but for challenging problems heavier smoothers can be useful. This plot is from my dissertation showing the solve time as a function of Poisson ratio for Jacobi, block Jacobi and "additive Schwarz" (overlapped). Its a little hard to read but at the highest Poisson ratio (0.499) overlapping Schwarz is about 2x faster than (vertex) Jocobi whereas it is about 3x slower at pedestrian Poisson ratios. 768