They're totally different objects. Unit balls encode distance by being the set of points within some distance from the center. They are thus, loosely, parameterized by a 1-dimensional measure.

Unit cubes are constrained such that all of their measurements are unit length---something which ensures that their volume is constantly 1. They do this by getting spiky, as xyzzyx mentioned. The distance to cross them shrinks as the vertex distance grows.

So they start to look totally different in high dimensions, evidenced by the ball's vanishing measure.