About ASP.NET Core

.NET Core includes console applications only, which is useful for various tools and local services. Additional application models have been built on top of .NET Core to extend its functionality, such as:

So, ASP.NET Core is built on .NET Core, but it also runs on the .NET Framework. See diagram.

ASP.NET 4.6 ASP.NET Core 1.0
.NET Framework 4.6 .NET Core 1.0
.NET Framework libraries .NET Core libraries
compilers and runtime components
.NET Platform (C#, VB.NET, F#, Roslyn, RyuJIT, SIMD)

Like .NET Core, ASP.NET Core can run on the platforms MacOS, Linux and Windows. It was designed to be leaner and more responsive than ASP.NET even on platforms other than Windows.