What makes the Poincarre Sphere different?

Answer 1

A Poincare Sphere is a manifold which is not homeomorphic to a sphere, but has the same homology as the sphere.

For instance, a tetrahedron wouldn't work as it is homeomorphic to a sphere. Almost anything we can imagine living in three dimensions will not work. See the source below for more info.
Bear in mind that in the source, it isn't the dodecahedron itself that is the Poincare sphere, only its image after suitable identifications have been made.

Answer 2

The Poincarre Sphere is different because it is not a perfect sphere, it is a perfect tetrahedron.