Nothing ... both do the same thing. "Classic" ASP is the first rendition of Microsoft’s version of Active Server Pages. ASP.Net is a total overhaul of the original concept allowing a larger separation of content and data. To the end-user, it’s all the same. To the programmer it’s the difference between haven and hell, with classic asp being hellish to maintain ... ;-)
There was a reason they called it spaghetti code.
Well, there are actaully differences, one being that asp.net is compiled code. Its correct that from the end user perspective, there is no difference, but that's true with any server-side scripting language since in all cases, what's sent back to the browser is HTML.
One thing to note is that it's not necessary to learn classic ASP before moving on to ASP.NET. If you are new to the Microsoft scripting, ASP.NET with C# is a good choice.
Not always compliled, compare a web site vs web application. But that is usually the big difference.
Another difference is mainatainability. New sites, if code in an OOP style, are much easier to maintain or modify.