All of the work in .NET Core benefits ".NET Native". .NET Native is what powers the .NET stack side for Universal Windows Platform (UWP) apps.

WinForms has been considered deprecated since WPF and WPF is essentially deprecated for UWP. (In an interesting historical aside, in some ways the UWP is what WPF was promised to be way back when WPF was still codenamed Avalon.)

Certainly WinForms and WPF aren't going anywhere any time soon (both in terms of backward compatibility will be stable for a long time still, but don't expect any new things), but UWP is clearly the way forward for new apps and it is benefiting from the work going into .NET Core.

The Desktop App Converter recently released (nee Project Centennial) can help if you've got a big investment in WinForms and wish to slowly migrate to UWP, so you aren't necessarily "stuck" in the past if you do want to try to move forward.

It supports UWP development. The .Net Framework isn't going anywhere though, if you're doing Windows desktop development in a legacy framework then you don't need the cross platform-ness of Core.

I could be totally wrong as I haven't really messed with it beyond dabbling, but from what I've heard, ASP.NET 4.? (6?) to ASP.NET Core is a pretty big shift which is why there's a lot of conversation around it. In fact, ASP.NET Core has a lot of changes vs the RCs (dipped toes into JSON configs, reverted back). Nuget has given up front end to bower now as well.

Perhaps that means desktop .NET Core is a more familiar experience? I'm not sure. But unless it is UWP or ASP.NET, I don't know how much we'll see coming as far as talk goes.

Honestly if I had to do a new desktop app project. I would probably go with nodejs + chromium + html (front end) & C# + .NET Core (backend). You get total control over the UI without having to deal with WPF, plus it runs cross platform.

Because Core is a server-side framework. WinForms and friends run on full framework.

That said, check out Avalonia and some other OSS experiments.

Microsoft is building their own new desktop apps on Electron. That tells you all you need to know.