Difference between 'Web Site' and 'Project' in Visual Studio

I have noticed that there is clearly a difference in what you get when you fire up Visual Studio 2008 and choose 'New Project' -> 'ASP.NET Web Application' instead of 'New Web Site' -> 'ASP.NET Web Site'. For example if you choose 'Project', then you can compile to .dll and each page gets a *.aspx.designer.cs codebehind file.

1) Why do we have these two different project types?

2) Which do you prefer?

3) Why would I choose one over the other?

4) What's the deal with the *.aspx.designer.cs files?