In general, a database solution will be most useful if you have a large number of features or need to filter or combine data. Do you want to display only polygons of a certain kind? Only polygons within 10 miles of a user-generated point? Do you need to know how much area each polygon covers? Or do you want to know how many features in another table share some attribute? Those are all good reasons to use a database where you can take advantage of the plethora of spatial functions and indexing options.
Even if you don't need to answer those kinds of questions, you still might consider storing your features in Postgres as polygons simply because it's nice to have all the data in one place and using Postgis is the "right way" to store spatial data in Postgres- especially if you are already using a Postgres DB for other data. The install should be fairly straightforward in any reasonably modern environment.