The main feature of Kurtowski's theorem on planar graphs at the time was that it connected two seemingly unrelated aspects of graphs, one topological in nature and the other combinatorial. Since then different authors have proved many theorems about planar graphs and they are now a fairly understood family. So I guess, the reasons for the importance of planarity have changed a bit with time.
Since many hard problems simplify a lot in the planar case this makes them a very good pedagogical tool to introduce when teaching/motivating various results in graph theory. As far as I know, planar graphs where the main reason to support Tutte's flow conjectures, for example. Other problems/conjectures where the planar case makes an interesting toy model are Fleischner's conjecture, circuit decomposition (Hajos), strong perfect graph conjecture, strong embedding conjecture, strong cycle double cover conjecture etc.
Part of motivation comes from topological graph theory, too. If you are interested in graphs as discretized version of surfaces, the planar case is probably the first you want to look into. Here the subject may jump to fields other than graph theory, though, and you may start to ask about properties under scalings and subdivisions (think of dimer models for example), but then there are more reasons that enter the picture for why the planar case is interesting, such as conformality for instance.