Honestly, .NET core makes a lot of sense for new projects that need to be highly scalable. If you have an API service that's hosted in a cloud provider, or a tool that generates PDFs of items, or a tool that scrapes API data and chucks it in a database, then this is great. These are simple applications that do not need an awful lot of authentication, crazy UI frameworks, or other windows-only features.
The code itself runs faster [1]. It might not matter for business / CRUD applications, but that's a nice to have thing.
The biggest selling point is cross platform support. Now, I don't care if the company switches from Windows 2016 VM's to some variation of RHEL. My code will work on that too. This gives your organization improved flexibility and reduced technical debt.
If you have a big legacy application using .NET Framework, it's probably not worth rewriting it for .NET Core. You will run into package conflicts. I have had to rewrite ".NET Framework only" pacakges to support .NET Core. That will throw your estimates off and probably delay the project. And that becomes really difficult to justify to project stakeholders.
[1]: https://www.ageofascent.com/2019/02/04/asp-net-core-saturati...
I tried 2-3 times up to and including .NET Standard 2.0 to switch and got burned by busted tooling and package conflicts every time. I’m starting a new project now and am still wary.
One of the major pains of my life as a .NET dev who does .NET core right now is Nuget and binding redirects!
This is why it is very interesting that the next "Core" release is to be called ".NET 5". No more "Framework", no more "Core". Simple version math applies: .NET 5 > .NET Framework v4.6.8 && .NET 5 > .NET Core 3.0. Maybe it will be less confusion all around, hopefully, soon.
Besides this future perspective, the real benefit from it you have today is that your project will (unless it uses Windows APIs, which is unfortunately way too common) instantly run on Linux.
Edit: can anyone give me the one-paragraph rational for why Net Core should take the world by storm? C# is a nice language and I'm a mostly happy VS user (except when I have to fiddle with MSBuild), but why should I go in and change the dropdown from ".Net Framework 4.5.2" to ".Net Core"? Let alone why any user of, say, Python or Node should drop everything and come over to C#.