Primary differences between Web Apps and Mobile Apps
A "web app" (short for web application) is any piece of software that runs through your web browser. Typically these are websites that provide some type of functionality. For instance SalesForce.com is a web app that lets you manage your various sales data tracking and reporting. Or you might open up your web browser to play a game online. These are web apps. Web apps are different then a website in the sense that a website is usually just information. For instance the site you are visiting right now (Switchbox's Website) is really just a website because it doesn't "do something."
A "mobile app" (short for mobile application) is basically the same thing but it runs on your phone or tablet and doesn't run through your mobile devices web browser. For instance, if you install SalesForce.com mobile application on your device you would click the icon and the application would open. If you opened up your web browser on your device and went to http://salesforce.com you would be accessing their web application, not their mobile application.
So who cares about web vs. mobile applications?
Generally speaking you want to build a mobile app if you need very specific device functionality. For security reasons web apps are not allowed to access you contacts or other specific device information. Mobile application can do so with your permission. Mobile applications also allow the application to use more features like your camera, microphone, accelerator, etc. Some web applications can be built to use your camera or microphone but these tend to be clunky. Generally speaking a mobile application will have better more robust functionality then a web application. Building a Native Web Application requires you to build a separate version for each device O/S (Android, Apple, Microsoft) but gives you the power to do more. Web applications are a bit more generic since they need to work across all device types.
So if mobile applications are better then web applications why doesn't everyone have one instead of a web app?
Mobile applications require more time to build and test. They also require you to submit your app to Apple, Android, Microsoft, Amazon or whomever you want to deploy your app to. Your users must also download your app which they typically won't do unless the functionality is there for them to do so. Mobile applications are also harder to maintain over the long term because each new device requires new testing and sometimes redevelopment. A web application is built to work with all devices (yes there are some issues across devices but these are small compared to mobile applications).
Example Mobile Application Built for a Columbus Ohio Company
Web Application Example