The Mosaic Cube is a vertex-turning cube puzzle. It is quite an interesting puzzle, and fun to play with, but it suffers from the same drawback that every vertex-turning cube puzzle must suffer from: it is too easy.
Why are vertex-turning cube puzzles so easy? In mathematical terms, it is because the axes of symmetry that pass through opposite vertices are not a generating set for the isometries of a cube. In simpler terms, you can’t get from one cube orientation to any other cube orientation simply by rotating around verticles.
In practical terms, this means that there the possible positions of a vertex-turning cube are much fewer than might be expected. Specifically, there is only one possible orientation for each edge piece in each position, so when you put an edge piece into its correct position, it must also be oriented correctly. This vastly simplifies the problem of solving it.
My approach to solving the mosaic cube starts with the realisation that it is impossible to separate neighbouring centre squares on advacent faces. Look at the solved cube above: see the yellow centre square nearest the green face, and the green centre square nearest the yellow face? No move can separate these. In fact, if you pull a mosaic cube apart, you’ll find that these two squares are physically connected — they are moulded out of a single piece of plastic. Based on that fact, I focus on placing edge pieces between their corresponding centre pairs. This is not hard to do, since there is no need to worry about orientation.
Once this has been achieved, the puzzle is reduced from a 4x4x4 Mosaic Cube to a 3x3x3 Mosaic Cube — does such a thing exist? — It is essentially Dino Cube with vertex pieces. Furthermore, note that around each vertex it is possible to rotate one layer or two layers, and only one-layer rotations break up our aligned edge pairs. In this reduced form, there is no longer any need to do one-layer rotations, and as long as we stick to two-layer rotations, we cannot undo the edge alignments we have done. Using two-layer rotations alone, it is not too hard to move our edge blocks around until the centres are solved. Thus the Mosaic Cube surrenders fairly easily to a reduction strategy.
As mentioned above, the Mosaic Cube is really just the big brother of the Dino Cube. It has an extra layer, some centre pieces, and some vertex pieces, but the fundamentals — in particular the vertex-turning nature of the puzzle — are the same. By the same token, it is somewhat related to the cuboctahedron-shaped Rainbow Cube, which is really just a Dino Cube with the vertices cut off.