Testing the application in intranet(withoutbrowser) is an example for client -server.(The company firewalls for the server are not open to outside world. Outside people cannot access the application.)So there will be limited number of people using that application.
Testing an application in internet(using browser) is called webtesting. The application which is accessable by numerous numbers around the world(World wide web.)
So testing web application, apart from the above said two testings there are many other testings to be done depending on the type of web application we are testing.
If it is a secured application (like banking site- we go for security testing etc.)
If it is a ecommerce testing application we go for Usability etc.. testings.
