It is increasingly difficult to find rental airplanes. After getting my PPL, Jeanene and I took a long driving trip through most of the states west of the Mississippi and turned into about every airport that we saw (mostly smaller ones because we avoided larger cities). At none of them could I rent an airplane right away. Either no rentals, rentals only to members of a flying club, rentals to those being trained, or a checkride was required and no CFI was available to give the checkride unless we waited until the next day or even the next week-end.
I don't think I save any by owning, but I do believe the cost is about the same as rental. So somewhere along the line toward or after PPL, it does make sense to purchase a plane. Its nice to have a plane available whenever I'm ready to fly, and I can go somewhere, park it, and not need to return it for the next renter.
And safety, too. I know the plane is in the same state at which I left it. Headphones, charts, radios, transponder, gas level, position of seats, etc., have not changed. That, along with getting to know the plane, has to make it safer to fly.