The difference between a shared project and a class library is that the latter is compiled and the unit of reuse is the assembly.
Whereas with the former, the unit of reuse is the source code, and the shared code is incorporated into each assembly that references the shared project.
This can be useful when you want to create separate assemblies that target specific platforms but still have code that should be shared.
See also here:
The shared project reference shows up under the References node in the Solution Explorer, but the code and assets in the shared project are treated as if they were files linked into the main project.
In previous versions of Visual Studio1, you could share source code between projects by Add -> Existing Item and then choosing to Link. But this was kind of clunky and each separate source file had to be selected individually. With the move to supporting multiple disparate platforms (iOS, Android, etc), they decided to make it easier to share source between projects by adding the concept of Shared Projects.
1 This question and my answer (up until now) suggest that Shared Projects was a new feature in Visual Studio 2015. In fact, they made their debut in Visual Studio 2013 Update 2
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