What is the purpose of websecurity.cs, if I can do this using the FormAuthentication and the Membership classes in System.Web.Security why would I need a class to do the same?
WebSecurity is developed for ASP.NET Web Pages. It is a simpler API for accessing the regular SimpleMembershipProvider.
You can use it to access the most frequently used security features. So it's not really a replacement for your FormsAuthentication and Membership classes. Instead it's a wrapper that simplifies development.
The MSDN Documentation shows how and when to use the class.