They do different things.
WebRequest is the abstract base class for HttpWebRequest – you can’t use it directly. It is the base class of other *Request classes (for FTP, File and other types of web requests). These classes are all used for getting resources (files) from the web.
There is also a WebClient class – probably the simplest to use of all the BCL classes designed to retrieve a resource from the Internet.
HttpRequest, on the other hand represents a request for a resource in an ASP.NET application – this is the server side of a Request.
The main difference is that HttpWebRequest is an HTTP client, and HttpRequest is server side to be used in an ASP.NET web application.