I am sorry for this “basic” question. But when I choose the type of one primitive I can choose between polygonal and mesh, and I always thought that they mean the same.

Not quite the same. Think of mesh as a kind of NURBS but with linear interpolation. So they are polygons but with a regular row & columns structure.
In many situations meshes are very useful (you can refine them in U and V, like NURBS), and there are also nice tools for them (PolyPatch for example). It's also useful that you can work with meshes then convert to polygons and get exactly the same representation.
Also check this in the docs:
hdox/houdini/content/base/polys.html