.NET team member here ... We like to think of .NET Native as a native tool-chain for .NET Core. I say "a" because the Xamarin and Unity (IL2CPP) tool-chains are equally applicable (and we've talked to them about it).
The core framework libraries (CoreFX) - https://github.com/dotnet/corefx - are used for all .NET Core scenarios, including .NET Native (UWP). This means that your code does the same thing in all of these different environments, since it's using the same underlying framework libraries. Separately, the Mono project is taking a lot of the same code, which means that the base framework for Xamarin apps are becoming more compatible with CoreFX, too. Yeahh! We hope to make this more formal in the future. We talk to @migueldeicaza about this frequently.
Today, .NET Native and .NET Core use two different runtimes, MRT and CoreCLR, respectively. MRT expands to the extremely creative "managed runtime". Colloquially, we call it "Mr. T". Earlier in the project, everyone working on .NET Native had posters of Mr. T (yes, that one) on our doors. Mohawks were entirely optional on the parts of team members.
MRT was built for static ahead-of-time (AOT) compilation. It is the child of the Redhawk project (http://www.zdnet.com/article/microsoft-codename-redhawk-live...). Redhawk was built to be a .NET-esque (using C# + extensions) systems programming environment. MRT was built to have many of the same benefits, but be compatible with .NET (not just "esque") and support nothing less, nothing more than C#.
CoreCLR is a child of the .NET Framework CLR. CLR (and by extention, CoreCLR) was built for dynamic execution, with a JIT. It supports AOT compilation in the pre-jit/NGEN sense.
The GC is the primary shared component between the two (MRT, CLR/CoreCLR) runtimes.
We plan to bring static compilation to more scenarios, however, we strongly believe that both JIT and (real) AOT are legitimate and compelling technologies and experiences and both are on our long-term roadmap. We'd like to leave the compilation choice/appraoch up to our users/customers. Novel idea, eh?
We have a lot of fun runtime and compiler tech on the team. It's a fun place to work. Much of it is now open source, meaning that we get to work on this fun tech in the open. This is our latest compiler announcement: http://blogs.msdn.com/b/dotnet/archive/2015/07/20/announcing....