The previous answers may describe the best understanding about the difference between .NET Core, .NET Standard and .NET Framework, so I just want to share my experience when choosing this over that.
In the project that you need to mix between .NET Framework, .NET Core and .NET Standard. For example, at the time we build the system with .NET Core 1.0, there is no support for Window Services hosting with .NET Core.
The next reason is we were using Active Report which doesn't support .NET Core.
So we want to build an infrastructure library that can be used for both .NET Core (ASP.NET Core) and Windows Service and Reporting (.NET Framework) -> That's why we chose .NET Standard for this kind of library. Choosing .NET standard means you need to carefully consider every class in the library should be simple and cross .NET (Core, Framework, and Standard).
Conclusion:
Microsoft just announced .NET 5: Introducing .NET 5