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