Benefit was vertice count for flat surfaces and better perspective accuracy sans-filtering.
PSone and Saturn didn't have perspective correction and used Affine mapping, but you probably only heard of warping on the PSone. There's several reasons for that, but the biggest is:
Affine is not good with triangles, with rectangles though, it behaves acceptably albeit not perspective correct.
Linear interpolation is the issue (it's a 2D texture manipulation, as you can see the squares don't get any smaller with distance, just warped, and the solution for both consoles was the same, subdivide it to make that "perspective" happen one way or the other, otherwise they could just do the surface of a fight ring in... say, Virtua Fighter II, to be a huge single quad and spare their polygon budget, or on the PSone they could make do with 2 polygons, and on both accounts it would look hideous. But more hideous on the PSone.
Then again, imagine you have to do the aforementioned flat surface with triangles, technically a quad is that very same polygon with one extra vertice going on whereas whilst working with triangles to do a rectangle you'll need to spend two of them. Depending on what you're doing the tradeoff might be good enough providing you have a limited amount of polygons to work with.
Anyway and in hindsight results on that front were way better by using quads and the cost was roughly 3 quads= 4 triangle polygons, tradeoff didn't feel wrong in the 90's, I guess.
One such recent example would be the Nintendo DS who used triangles and quads, unlike most systems where if you want to you can run the system into the ground with geometry, it just won't be realtime anymore (...) the DS was actually geometry capped to 2048 polygons or 6144 vertices, per frame.
The thing could reach these numbers at 60 fps with free 2xAA, it was just capped (probably due to RAM budget).
Thing is, it supported quads. So these 2048 polygons could amount to as much as double 4096 polygons if they were all quads and you were to use triads instead. It sure came in handy, obviously, counting a rectangle as 4 vertices and not 6 could really mean something in the long run. (I believe geometry had to be optimized for that though, and most devs wouldn't.