Look at Google Docs, which has changed the way teams collaborate. Look at Unity which recently ported their player to the browser. Look at the Autodesk 123D suite.
With technologies like WebGL and Emscripten, we're just starting to see the tip of the "native on web" iceberg. And it's awesome. I know because I'm helping build the iceberg... :)
It's awesome on the desktop though, but there are frustratingly good reasons that Google couldn't just run that code on phones and expect it to be anything other than a terrible experience.
Not in my experience. I've never had a native app pop-up a dialog saying "You can't use that menu to copy and paste, you have to use the keyboard shortcut." Instead, the standard platform way of doing things just works. Then there is the fact that Google Docs just refuses to open certain docs in certain browsers.
It's impressive for something on the web, but it's not better than a native app by any stretch.
But Google Sheets has transformed many collaborative functions in our organisation. Especially at budget time.
The ability to have concurrent access to the latest version always is wonderful. No more emailing attachments.
Also being able to sit down at a budget meeting and everyone can view the work-in-progress as it develops on their own device.
In all of this we've never had any one bothered by lack of speed.