Roughly, a cubical complex is like a simplicial complex except all the pieces glued together are combinatorial cubes of various dimensions. A cubical sphere is a cubical complex that is homeomorphic to a sphere. I have encountered papers that distinguish between cubical spheres and cubical polytopes, but I do not understand the distinction. Is there a distinction already in $\mathbb{R}^3$? If so, could anyone provide an example? A reference to clear definitions would suffice as well. Thanks!
My understanding is that, say, the rhombic triacontahedron is both a cubical polytope and a cubical sphere in $\mathbb{R}^3$:
Image from Wikipedia article
(Too long for a comment): Here are some thoughts on this circle of ideas though I am not sure what distinctions have been made with these terms in the literature. The diagram you show is a 3-polytope whose surface is built up of 2-dimensional combinatorial cubes, namely 4--gons. However, it is not clear that this 3-polytope or similar 3-polytopes including their interior points can be always be decomposed as 3-cubes that meet along faces. The note on this page which talks about combinatorial cubes: http://www.york.cuny.edu/~malk/tidbits/n-cube-tidbit.html shows a diagram of as 4-cube but it also can be thought of a 3-cube whose interior has been cut up into other combinatorial 3-cubes. As regards the torus, there are some graphs which will not embed on a sphere but will embed on a torus. Now one can ask if one can embed in 3-space a surface (topologically a torus) with flat faces so that the vertex edge graph of this surface is the given graph. When this is possible it is common to call the resulting surface a toroidal polytope. The adjective toroidal overcomes the usage of polytope to be something convex.