For me, the answer is "because native mail clients are so much snappier and responsive".
As well as being able to choose my client based on the interface I prefer, rather than whatever UI the webmail provider decides to give me. And my mail client is also always open. And it uses less memory than a typical browser tab.
The reality I think is, the average user probably doesn't care about snappyness as long as it is not too bothersome. The path of least resistance is opening the webmail (no setup required).
