For a web app to provide the same functionality as a native mobile app the same content must be downloaded. A mobile app downloads compiled executable and all assets in a bundle upfront, then it's permanently stored. A web app downloads UI structure (HTML,CSS) and minimal code (JS) first, appears to instantly be working, then downloads the rest (media assets like images and movies, the rest of the code, etc) in the background. In the end, the same amount of stuff has been downloaded. But web apps have the instant usability feel to them.
Web apps are improving such that they will be able to match the same experience and speed of native apps. JS engines run faster, WebGL allows 3D games, caching of files allows faster starting times. Maybe they'll get close to native speed, unlikely to match it due to middle man (browser, plus overhead of compiling JS vs pre-compiled Obj-C). But you still have re-downloading when the cache clears something out (or else you copy more mobile app features by letting the user declare what stays in the cache permanently). Mobile apps are re-downloaded once per update. Web apps will be re-downloaded more often than that even with good caching (maybe, I guess you could cache infinitely until you run out of space, so with sufficient space it's equivalent). So while web apps are playing catchup quite well, mobile apps still have advantages.
Payment models are different generally, but the models would work fine regardless of the web app vs. mobile difference. Web apps tend to be subscription based, mobile apps tend to be buy once with infinite free upgrades, or using some newer experimental model like "freemium". But you could make a buy-once web app or a subscription mobile app, and IIRC people already have.
So, the technologies are converging, and at the end of the day you still have to get the software from the developer to the user.
JS+WebGL+CSS+HTML+localstorage+... vs. Obj-C+Cocoa, that's just format war. Open web+browsers vs. curated Apple App Store + iOS vs. un-curated Google App Store + Android, that's platform war. Someone will win. But people will be using software on their mobile devices, regardless of how and where they get it or what's under the hood. Mobile is next, and the interconnectivity of the web will be working with it, and social will be working with it, and whatever else comes in the future will also add and change things of course.