while triad-based chords imply specific harmonies, quartal chords are more generic and basically don't imply specific harmony in the key you're in, which means you can use them pretty much anyplace and in lieu of regular chord progressions. Quartals are more vague so they fit in a wide variety of situations.
Quartals can also be thought of as voicings for extended chords (9ths, 11ths, 13ths) and so therefore can be used as ways of voicing triads when you want to use triads with their tensions (extended notes).
A good explanation of this is in the short booklet: Jazz Piano Comping: intermediate, by Tom Anderson.
A good real-life illustration of how these chords are used on piano is found in any of the Jamey Aebersold "Piano Comping" and "Transcribed Piano Voicings" volumes. These are the transcriptions of the pianists who are comping along Aersold's play-along recordings. When you look at what they're doing, you see that they use quartals a lot and they use them very freely regardless of what the written set of chord changes is supposed to be.