MVC 6 was part of ASP.NET 5, but due to some major changes in the code base, they decided to change its name from ASP.NET 5 to ASP.NET Core.
You can read about here: http://www.hanselman.com/blog/ASPNET5IsDeadIntroducingASPNETCore10AndNETCore10.aspx
The main difference between ASP.NET and ASP.NET Core are as described below…
Platform Independent Features ASP.NET Core provides hosting in multiple platform like Linux or Mac or Windows hosting (Cross platform).
Introduced new lighter version of assemblies with non dependent frameworks.
Ready for cross platform cloud hosting.
Frameworks are a complete rewrite (and still building)
Project default project structure got renewed with type scripts configurations.
Improved new security features and dependency inversions.
Please visit .NET Core and ASP.NET Core for more details
The main reason behind the core framework is dependency inversion. As its framework is completely rewritten in such a way that high level assemblies completely decoupled from low level supporting assemblies.