.NET Core allows side-by-side installation, so different applications can use different .NET Core versions.
.NET Core is better suited for
microservices
Docker containers
high performance and scalable systems
.NET Framework might have
third party .NET libraries or NuGet packages that are not available for .NET Core
.NET Framework technologies that are not available in .NET Core
ASP.NET Web Forms applications
ASP.NET Web Pages applications
WCF service implementation (might be added in a future version of .NET Core).
Windows Workflow Foundation (WF), Workflow Services (WCF + WF), WCF Data Services (formerly known as ADO.NET Data Services)
.NET Framework is updated with Windows Update, .NET Core with its package manager NuGet.
.NET Framework implements the Common Language Interface (CLI) with the CLR and Framework Class Library, .NET Core with CoreCLR and CoreFX.
.NET Framework will be deprecated and the release after .NET Core 3 will be .NET 5 (without the suffix Core), to avoid confusion with the .NET Framework.