Website:

The Web Site project is compiled on the fly. You end up with a lot more DLL files, which can be a pain. It also gives problems when you have pages or controls in one directory that need to reference pages and controls in another directory since the other directory may not be compiled into the code yet. Another problem can be in publishing.

If Visual Studio isn't told to re-use the same names constantly, it will come up with new names for the DLL files generated by pages all the time. That can lead to having several close copies of DLL files containing the same class name, which will generate plenty of errors. The Web Site project was introduced with Visual Studio 2005, but it has turned out not to be popular.

Web Application:

The Web Application Project was created as an add-in and now exists as part of SP 1 for Visual Studio 2005. The main differences are the Web Application Project was designed to work similarly to the Web projects that shipped with Visual Studio 2003. It will compile the application into a single DLL file at build time. To update the project, it must be recompiled and the DLL file published for changes to occur.

Another nice feature of the Web Application project is it's much easier to exclude files from the project view. In the Web Site project, each file that you exclude is renamed with an excluded keyword in the filename. In the Web Application Project, the project just keeps track of which files to include/exclude from the project view without renaming them, making things much tidier.

Reference

The article ASP.NET 2.0 - Web Site vs Web Application project also gives reasons on why to use one and not the other. Here is an excerpt of it:

• You need to migrate large Visual Studio .NET 2003 applications to VS 2005? use the Web Application project.
• You want to open and edit any directory as a Web project without creating a project file? use Web Site project.
• You need to add pre-build and post-build steps during compilation? use Web Application project.
• You need to build a Web application using multiple Web projects? use the Web Application project.
• You want to generate one assembly for each page? use the Web Site project.
• You prefer dynamic compilation and working on pages without building entire site on each page view? use Web Site project.
• You prefer single-page code model to code-behind model? use Web Site project.

Web Application Projects versus Web Site Projects (MSDN) explains the differences between the web site and web application projects. Also, it discusses the configuration to be made in Visual Studio.

Yes, the Session object might be null, but only in certain circumstances, which you will only rarely run into:

• If you have disabled the SessionState http module, disabling sessions altogether
• If your code runs before the HttpApplication.AcquireRequestState event.
• Your code runs in an IHttpHandler, that does not specify either the IRequiresSessionState or IReadOnlySessionState interface.

If you only have code in pages, you won't run into this. Most of my ASP .NET code uses Session without checking for null repeatedly. It is, however, something to think about if you are developing an IHttpModule or otherwise is down in the grittier details of ASP .NET.

Edit

In answer to the comment: Whether or not session state is available depends on whether the AcquireRequestState event has run for the request. This is where the session state module does it's work by reading the session cookie and finding the appropiate set of session variables for you.

AcquireRequestState runs before control is handed to your Page. So if you are calling other functionality, including static classes, from your page, you should be fine.

If you have some classes doing initialization logic during startup, for example on the Application_Start event or by using a static constructor, Session state might not be available. It all boils down to whether there is a current request and AcquireRequestState has been run.

Also, should the client have disabled cookies, the Session object will still be available - but on the next request, the user will return with a new empty Session. This is because the client is given a Session statebag if he does not have one already. If the client does not transport the session cookie, we have no way of identifying the client as the same, so he will be handed a new session again and again.