Comparing ASP.NET to C#.NET is like the proverbial apples to oranges comparison.
ASP.NET is a web application framework built on top of the .NET framework (In much the same way as Ruby on Rails is to Ruby, or JSP is to Java). It can use many (if not any) of the .NET framework languages. ASP.NET and C#.NET are not mutually exclusive. I have built web applications, console applications, windows services, and windows forms applications in VB
.NET, C#.NET and J#.NET.
If you find a job listing that consider ASP.NET and C#.NET to be "different" then don't consider taking it. They clearly don't know what they are talking about. I can only assume that these kinds of people and companies consider "C#.NET" to be windows forms programming.