Xamarin Studio didn't support .NET Core, so previously you had to use the CLI tools and an editor like VS Code. You could run ASP.NET MVC in XS (version 5, Core is effectively version 6) but this ran on Mono.

Visual Studio for Mac (aka XS vNext) does now support ASP.NET Core so there is less need for VS Code. The re-brand makes sense (it's not just for Xamarin now). However, Code runs on more platforms (Linux, ARM etc.) and is also used in the browser as part of Azure.

I wrote a better summary of the differences here: https://unop.uk/getting-started-with-c-sharp-and-cross-platf.... It's the last in a four part blog series about .NET on Mac.