Why do we care about planar graphs? Planar graphs are graphs that can be embedded in the plane. Classic examples of planar graphs are the $1$-skeleton (vertices and edges) of polyhedrons.
Most introductory books on graph theory will have a chapter on planar graphs. Why are planar graphs important, besides the characterization of planar graphs being a historical question?
Does knowing that a graph is planar (versus knowing its genus) bring important tools to study it?
 A: In "Pure Math" terms , Planar graphs are interesting objects in themselves. Mathematicians ( like Kuratowski & Wagner & Euler ) analyse those objects to extract interesting theorems.
There are many outstanding conjectures among Planar graphs ( like those by Harborth & Scheinerman & Barnette ) hence more work will continue in that Area.
In "Applied Math" , general graphs are everywhere. In Particular , Planar graphs are necessary in specialized cases :

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*Designing Printed Circuit Boards , MicroProcessors , where the wires are on a Plane.

*Roads & Railway tracks , where crossings will block the traffic or will necessitate costly bridges.

*Irrigation Canals & waterways , where Crossings are not admitted.

*Visualizing large Interconnected temporal or geographical Data , where Crossings will hurt human comprehension.

*Chemistry & Quantum Physics , where the atoms or Particles are connected & Crossings are Prohibited in certain cases.

*Cartographical & Political land-masses are generally Planar.

This Article Discussing Planarity was generally useful.
There are other online Article too.
