The mesh to the right was changed, it was decimated, simplified, some kind grid parameterization, Poisson ???, Laplace ???, (don't knw the correct names). PropViewer doesn't change the number of vertices and its position, neither UV. It only combines two triangles into a quad where is possible.

MeshLab has a lot of tools for editing and correcting meshes or scanned range data, sometimes crashes and the output is always a triangle mesh. This doesn't matter because once you have a regular grid of triangles in the way you want you can convert into quads using PropViewer

Stupidity also evolves!