the first allow mocking and testing any PHP function. This part is a general tool and two times framework agnostic: can be used to test code that uses any frameworks (or no framework) and in combination with any testing framework.
the second group of features can be used with any testing framework as well, but is specific to test WordPress code.
Who is interested in the first part can use only it, just like this second group of features does not exists.
Why Brain Monkey
When unit tests are done in the right way, the SUT (System Under Test) must be tested in isolation.
Long story short, it means that any external code used in the SUT must be assumed as perfectly working.
This is a key concept in unit tests.
In PHP, to create "mock" and "stubs" for objects is a pretty easy task, framework like PHPUnit or phpspec have embedded features to do that, and libraries like Mockery make it even easier.
But when external code make use of functions things become harder, because PHP testing framework can't mock or monkey patch functions.
This is where Brain Monkey comes into play: its aim is to bring that easiness to function testing.
define functions if not defined
allow to enforce function behavior
allow to set expectations on function execution
Moreover, I have to admit that I coded Brain Monkey to test WordPress code (that makes a large use of global functions).
This is the reason why Brain Monkey comes with a set of WordPress-specific tools, but the ability to monkey patch and test functions is independent from WordPress-specific tools and can be used to test any PHP code.
Under the hood
Brain Monkey gets all its power from two great libraries: Mockery and Patchwork.
What actually Brain Monkey does is to connect the function redefinition feature of Patchwork with the powerful testing mechanism and DSL provided by Mockery, and thanks to that Brain Monkey has:
PHPUnit, PHPSpec or any other testing framework compatibility
powerful and succinct API with human readable syntax