Distinction between this geometric example of a Diffeomorphism & a Homeomorphism

[damnedcat]

when I first learned about homeomorphic sets, I was given the example of a doughnut and a coffee cup as being homeomorphic since they could be continuously deformed into each other. fair enough.
Recently I heard another such example being given about diffeomorphisms: "Take a rubber cube. Loosly, Anything you can do to this cube without tearing it or glueing two parts together is a diffeomorphism. You can stretch it, rotate it, twist it, or do each of these things at different points. In other words it's the most general kind of transformation of the cube that is "nice" in some sense."

Now these two geometric examples seem to be essentially the same thing, Am I missing some subtle difference between these examples?

Do the apparent similarities in the geometric examples lie in the fact that homeomorphic surfaces are diffeomorphic in 1,2 or 3 dimensions?
(without getting overly techical and sticking to the geometric interpretation)

 

[Ben Niehoff]

A homeomorphism need only be continuous; whereas a diffeomorphism needs to be differentiable. So for example, changing a cube into a sphere is a homeomorphism, but not a diffeomorphism, because the cube has corners that you can't eliminate in a differentiable way.