I presume your ASP.NET web site is using Windows Integrated (NTLM) authentication. A user authenticated this way cannot authenticate to a second location from the server side (the web server.) You are experiencing what is known as the "double-hop" (1) limitation of NTLM. You must use a dedicated account on the server side, or if you really do want to use the logged-in user's identity, you must use an authentication scheme that permits delegation, such as Kerberos.
If you really need the user's identity to access SharePoint data and you cannot change the authentication scheme, then the best way to do this is to use the JavaScript CSOM. This means the user is authenticating directly to the SharePoint server (a single hop, not double) and your ASP.NET site serves the page containing this script to the user.
(1) http://blogs.msdn.com/b/knowledgecast/archive/2007/01/31/the-double-hop-problem.aspx