It's faster and can be miniaturised more than analog ic's, possibly using less energy. Version used in article is proof of concept, selected for being easy to manufacture with cheap equipment.

I don’t know much about analog computing, but how well can analog computers solve PDE’s in 3 spatial dimensions?

I know that until digital computers came along, structural analysis was done with strains lacquer techniques, so the circuit way seems limited

The advantages won't be clear until they do more testing, but it might be possible to be more power efficient that silicon ASICs.