Classic mode models the IIS 6.0 model in which ASP.NET is an ISAPI add-on to IIS. This mode is available for backward compatibility but lacks many of the features in the new Integrated mode. In Classic mode, IIS has its own pipeline that can only be extended by creating an ISAPI extension, which has a well-deserved reputation for being difficult to develop. ASP.Net is run as an ISAPI extension that is just one part of the IIS pipeline.
Integrated mode makes ASP.Net an integral part of IIS. Now the IIS server functionality is split into more than 40 modules that break the IIS and ASP.Net functionality into pieces. Modules such as StaticFileModule, BasicAuthenticationModule, FormsAuthentication, Session, Profile and RoleManager are part of the IIS pipeline. FormsAuthentication, Session, Profile and RoleManager were previously part of ASP.Net and didn't have anything to do with IIS.