".NET is a developer platform made up of tools, programming languages, and libraries for building many different types of applications.
ASP.NET extends the .NET developer platform with tools and libraries specifically for building web apps."
I've been developing with ASP since before .Net. In the late 90's, Microsoft released Active Server Pages as a way of scripting database applications. When .Net was introduced, ASP became ASP.Net. Many applications had already been written in .Net, so I wrote and marketed a utility called SessionBridge to help deal with a breaking change that would hinder developers from porting their applications from ASP to ASP.Net.
When ASP.Net was first released, it used "WebForms" which allowed developers used to writing desktop applications transition to the Web. Years later, ASP.Net MVC (Model-View-Controller) was released and quickly became the standard approach to developing ASP.Net applications.
Now Microsoft has released the next generation - ASP.Net Core, which allows applications to run on Linux and Macs as well as Windows.
I've written countless ASP and ASP.Net applications and have now transitioned to developing for ASP.Net Core. All of my new development is ASP.Net Core and I hope to have my existing ASP.Net based products converted to the new platform in the next few months.