The pros of native app development are primarily around getting access to hardware features that aren't accessible through web APIs, obtaining native performance benefits (such as in action gaming), instant access to paying customers through a platform store (such as iTunes), and security situations where you don't trust the browser or how it's handled.
The cons of native app development are that you lock yourself into a potentially proprietary code platform, write a bunch of device-specific code, and you're vendor locked. Code is harder to write, much harder to deploy, and you stand the chance of having the rug pulled out from under you. (Yes I'm looking at Apple, but could happen to any proprietary platform.)
Web apps by contrast are based on technologies that are widely known and easy to deal with - HTML5, CSS3, JavaScript, and excellent libraries such as JQTouch are available to help. Well-designed web apps for the most part will not care if you're on a Blackberry, Android, or iPhone, and will work on many of the older and less capable models as well as the newer ones and devices we haven't even encountered yet, without having to recompile or refactor (or at least without having to do a great deal of recompiling or refactoring...) And there are some hardware features accessible, such as GPS through the geolocation API.
But on the other hand web apps may not perform well with large data sets or high computational requirements. If you're building a commercial app with financial transactions, you very likely will have to roll your own payment system. And you have to trust the browser security as well.
All in all, most apps are going to make the best sense as web apps. However, many web apps can be made to function to be almost indistinguishable from client apps. With some HTML5 offline storage, CSS3 and JS functionality for transitions and behaviors, many business apps can be made to be indistinguishable from native clients.
In iPhone's case we can take it further: Adding a 57x57px icon apple-touch-icon.png to your web app’s root directory will provide iPhones with a nice custom icon when users add an app to their home screens (iPhone will take care of the rounded corners and glossy visual effect) and you can make an iPhone app go full screen when clicked from it’s home screen icon by adding . At this point, the app has it's own icon and runs full screen - the user doesn't know it's web based.
And if you do want to go native but don't want to abandon web standards, most native APIs provide the ability to develop native clients based on HTML/CSS/JS using a simple wrapper, such as the UIWebView in Objective-C. PhoneGap is an excellent cross-platform framework enabling standards-based web development practices to be deployed on iPhone, Android, and Blackberry.