We use .NET Core for our web socket servers, mail relays, and scheduling applications. Soon, it will power all of Stack Overflow. The ability to run on any platform, further extend and plug especially the ASP.NET bits and treat almost everything as a building block you can move around has been a huge win. We're headed towards an appliance model and with .NET Core we can finally put everything in box...on Linux. We can re-use more code, fit all our deployment scenarios both during the move and after, and also ditch a lot of performance workarounds we had to scale...they're in-box now.
And testing. The ability to fire up a web server and request and access both in a single method is an orders of magnitude improvement over ASP.NET 5. We're looking forward to tremendously improving our automated test coverage in places it's finally reasonable in both time and effort for devs to do so. In short: we're getting a lot more for the same dev time spent in .NET Core.