Omar, I don’t know, I think they look similar in a lot of cases, but there are situations that they can look completely different.
As far as I understand, with Harden Normals, Blender looks at the neighbouring Faces and with the Weighted Normals (which I have never used yet, to be honest) it also takes into account, for instance (by default) the size (area) of those Faces.
So, Harden Normals is like an on/off switch, while with the Weighted Normals you get a lot more control.
It’s a bit vague description, but that is because I don’t really know.
