In graph theorie nodes and edges are "dual". Whenever you have a graph with weighted nodes you can construct a corresponding "dual" graph with weighted edges, and vice versa. And all algorithms are equally transformable.

So "why weighting nodes" has the same legitimation as asking "why weighting edges".

It's just a matter of perspective.

PS: Wikipedia restricts this to planar graphs !?! I'm too lazy to verify what normally happens after transposing the incidence matrix... it certainly works for lattices and they are seldom planar graphs.

UPDATE: OK the duality for lattices is very different, from the graph perspective it's just mirroring at the horizontal axis. I'm quote rosted in this 8(

Anyway at least for planar graphs weighting can be "dualized"!