JQuinn - offline access would be a big one right now (although html5 technologies are catching up with native). But, try this on your device - goto settings, and put it in airplane mode (this takes it offline).
Now, let's say you are needing to look up an appointment on a calendar. Try and open up Safari to get to your google calendar. You won't be able to do it because you do not have a connection. Now, open up the calendar app (which is native). You will be able to see all your appointments, create new appointments, etc. without being connected.
That is just one of the main differences between native and web apps.
A good article to read about native vs. web apps is here:
Web Apps vs. Native Apps
That one is pretty good at a brief description of what can and cannot be done. directly from Safari, Photos, Camera, YouTube, and Maps
The new iOS 5 twitter integration is a great example (if you haven't upgraded your iPhone software for a while and can get the latest version you definitely should do it
The new iOS 5 software that Apple released a few months back includes a really cool twitter integration feature which allows you to tweet from just about anywhere directly from Safari, Photos, Camera, YouTube, and Maps, etc. Now, Apple also released this feature to developers when they released the latest SDK (software development kit) to us. So, now, I am going to be including this cool little twitter functionality within our App here pretty soon. This would not be available to me if I had developed a web app which is accessed through Safari.
Hope this all makes sense.

Please feel free to ask more questions on this or if I can provide a better explanation.
Here is another good article about it:
Sam Soffes - Web App vs Native App
By the way, don't get me wrong - I am not "anti-web app". I love the HTML5 technology and where it's going. I wish we could have used it to build our app. But, the fact is, it is just not there yet when it comes to speed and access to the devices functionality. I think it will get there some day. But, for right now, native it definitely the way to go.