ASP has always used other languages to code its pages. ASP stands for Active Server Pages, which means you can have dynamic content based on a server program. So, it's technically a name for the technology and not a language, just as DHTML (Dynamic HTML) is actually a mix of HTML and Javascript, and not a language in its own right.
ASP, according to Wikipedia, was able to use many languages, such as JScript, VBScript, and even PerlScript.
ASP.NET takes Classic ASP further. It allows you to use the .NET Framework within your web application. The best part is that the same languages can be used to develop desktop applications, or interfacing web services, etc. So now you can do pretty much anything with knowing only a single language, though you do need to know how web applications differ from desktop apps, services, etc.
ASP.NET does have its own approach to programming, such as different events that fire during different sections of a page's life cycle. This is where you need to know ASP.NET-- if you are unfamiliar with how to approach the structure of a web application, you'll definitely run into some problems down the road.
When an employer says 'ASP.NET, VB.net experience required,' they are generally referring to you being able to program ASP.NET web applications in VB.NET, as ASP.NET can also be programmed in C#.NET. On the other hand, they could also mean that you will be building desktop apps as well, using VB.NET. Thanks to the .NET Framework, both are similar, but as I said before-- Just because you know VB.Net doesn't mean you know ASP.NET.