In Visual Studio Enterprise 2015 (v14.0.25123.00 Update 2) when I go to:
Start > New Project
With Target Framework 4.5 or higher selected...
Under:<
There is .Net Core and .Net Framework. .Net Framework only works in windows environment while .Net Core is an open source framework that runs on windows, linux and ios. .Net Core offers essential functionalities (everything you need, including database communication) for web applications, while .Net Framework contains functionalities that tie with windows.
ASP.NET Web Application works on .Net Framework 4.6.1 and older, while ASP.NET core Web Application works on .Net Core and/or .Net Framework 5, which is why there are 2 ASP.NET core web applications.
If your aim is cross platform, or not sure, then start with the .NET core version. If your aim is on MS OS, then use .Net Framework version.
The only difference is the reference within the project, the rest is identical.