This kind of mesh is used quite frequently in fields where the goal is to simulate a thin spherical shell and if the goal is to use quadrilateral cells (or hexahedral if the third dimension/depth/height is relevant). Atmospheric modelers use it for the atmosphere, but it is also frequently used in simulating convection in the Earth mantle, for example. (You can't see the coarse mesh, but this simulation uses something similar to the cubed sphere mesh extruded into the depth of the Earth: https://www.youtube.com/watch?v=j63MkEc0RRw .)
If your code is based on triangles, then starting with a icosahedron would work just as well. Using quadrilateral meshes like for the cubed sphere has the advantage that one can use 6 nicely structured rectangular meshes for which the transformation from reference cell to real cell is easy to write down -- in essence, one works on 6 rectangles subdivided into orthogonal meshes. Whether you like quadrilateral or triangular meshes better is large a matter of choice and taste.