Not a lot. One uses the Operating System interface to display a rich user interface, the other uses the browser and usually delivers a web service (like search).
The Mobile app gets access to all of the capabilities of the device via the Operating System API's (application programming interface) whereas the the Web app has to make do with the browser and JavaScript.
The other big difference is that it's much harder to build cross platform Mobile apps due to all the different operating systems than it is to build a web app that displays content in the browser.
Here's a simple scenario. You decide to build a Mobile app that does Search - it will have a nice user interface and connect in the background to your web service to deliver the results. All the user will ever see is the Mobile app interface. You decide to deliver this app on 6 Mobile platforms - here's what you have to learn:
- Blackberry - their version of Java
- Symbian - J2ME (standard version of Java)
- Windows Mobile - C++
- Android - their version of Java
- Apple - Objective C
- RIM - WebOs
6 different platforms - 6 different operating systems - 6 different form factors - 6 different interfaces - and none of the code is usable on the different platforms, essentially requiring you to support 6 different code bases.
Now imagine that after releasing the Mobile app you need to add 4 new features. Ouch!
Compare this with a Web app inside the browser - all of the above devices have a standard browser that displays HTML and supports Javascript. Essentially you can write once and "display" everywhere, even on the desktop.
The cost, development time and risk are considerably diminished with a Web app. Mobile apps only make sense in a niche, one platform environment - otherwise they simply don't scale from a cost and risk perspective.
So why do people build Mobile apps? Simply because they can access the device side capabilities that the Web app/browser can't.
However that's now changed and it's possible to access any device side capability via the browser without compromising security or performance - there's a free trial version out there that shows you how it's done. (link).