ASP.NET Core 1.0 is new branding name for ASP.NET 5 which is a new open-source and cross-platform framework for building modern cloud-based web app. .NET Core 1.0 on the other hand is the new .NET Core 5 which is a refactored version of .NET. It was redesigned to be modular which allows developers to plug components which are only required for your project, most features will be available as plugins via NuGet. Microsoft rebranded the previous ASP.NET 5 stuff because the whole .NET now is new that's why the Core 1.0 was born.
.Net Core 1.0 is to provide cross platform web app development and ASP.NET Core 1.0 is totally new open source ASP.NET with lot many great features.
Both are like open source technology by microsoft
Because these are new. The whole .NET Core concept is new. The .NET Core 1.0 CLI is very new. Not only that, but .NET Core isn't as complete as the full .NET Framework 4.6.
.NET Core is a cross-platform, open source, and modular .NET platform for creating modern web apps, microservices, libraries and console applications.
Since ASP.NET Core is now framework independent, it can be hosted outside the IIS. You can use OWIN to wrap your application and host it whenever you want for example Linux
If you develop an application using ASP.NET Core.. you can deploy the applicaitons in linux and Mac operating systems as well..
ASP.NET 5 is now ASP.NET Core 1.0. .NET Core 5 is now .NET Core 1.0. Entity Framework 7 is now Entity Framework Core 1.0 or EF Core 1.0 colloquially.
ASP.NET 5 is now ASP.NET Core 1.0. .NET Core 5 is now .NET Core 1.0.
.NET Core 5 is a modular runtime and library implementation that includes a subset of the .NET Framework. Currently it is feature complete on Windows, and in-progress builds exist for both Linux and OS X. .NET Core consists of a set of libraries, called “CoreFX”, and a small, optimized runtime, called “CoreCLR”. .NET Core is open-source, so you can follow progress on the project and contribute to it on GitHub
Main Feature of ASP.NET Core 1.0 is cross platform development
Why 1.0? Because these are new. The whole .NET Core concept is new. The .NET Core 1.0 CLI is very new. Not only that, but .NET Core isn't as complete as the full .ASP.NET Core 1.0 is a 1.0 release that includes Web API and MVC but doesn't yet have SignalR or Web Pages. It doesn't yet support VB or F#. more details check below link : http://www.hanselman.com/blog/ASPNET5IsDeadIntroducingASPNETCore10AndNETCore10.aspx