I'm not saying that you wouldn't be able to do this with a web-based mail client, but I think that the machinery to do that would be less natural. (My sense of what is natural might be warped, though.)
Also, there is much to be said about the ability to customize your mail client to suit your needs. The very fact that people are complaining about Gmail's new design shows that a lot of people are relying on Google to tell them how they should manage their email, and try to adapt their needs to the tool rather than the other way around. This is not a technical limitation of web-based mail clients, but most webmails I know aren't really hackable.