What are the differences between Visual Studio Code and Visual Studio?

Visual Studio (full version) is a "full-featured" and "convenient" development environment.

Visual Studio (free "Express" versions - only until 2017) are feature-centered and simplified versions of the full version. Feature-centered meaning that there are different versions (Visual Studio Web Developer, Visual Studio C#, etc.) depending on your goal.

Visual Studio (free Community edition - since 2015) is a simplified version of the full version and replaces the separated express editions used before 2015.

Visual Studio Code (VSCode) is a cross-platform (Linux, Mac OS, Windows) editor that can be extended with plugins to your needs.

For example, if you want to create an ASP.NET application using Visual Studio Code you need to perform several steps on your own to setup the project. There is a separate tutorial for each OS.

Visual Studio Code is an editor while Visual Studio is an IDE.

Visual Studio Code is cross-platform and fast, while Visual Studio is not fast.

Note that Visual Studio for Mac is available now but is a different product compared to Visual Studio (Windows). It's based on Xamarin Studio and lacks support for some older .NET project types. It does successfully build solutions created in Visual Studio 2017. Visual Studio for Mac has a more limited UI (for example, no customizable toolbar). So for cross-platform work, Visual Studio Code may still be preferable.