There are an infinite number of regular polygons, but only a finite number of regular polyhedra. Given two equal-area polygons, one can be dissected into a finite number of pieces, then reassembled into the other. This is usually not true for pairs of equal-volume polyhedra. There exists a square that is decomposable into smaller, different, squares. There is no cube that is decomposable into smaller, different, cubes.
All polygons are decomposable into triangles by adding only interior edges.
Not all polyhedra are decomposable into tetrahedra by adding only interior faces. For polygons, all such decompositions have the same number of triangles. Some polyhedra can be decomposed different ways into different numbers of tetrahedra. Every polygon has every interior point visible from some vertex. Some polyhedra have interior points not visible from any vertex.