A black box test (also known as a "functional test") is one in which you feed it inputs and verify the outputs without being able to inspect the internal workings.  Furthermore, one doesn't usually have information regarding:

• how the box handles errors
• whether your inputs are executing all code pathways
• how to modify your inputs so that all code pathways are executed
• dependencies on other resources

Black box testing limits your ability to thoroughly test the code, primarily because the you don't know if you're testing all the code pathways.  Typically, a black box test only verifies that good inputs result in good outputs (hence the term "functional test").

Classes are often implemented as black boxes, giving the "user" of the class access only to the public methods and properties that the implementer selected.

A white box provides the information necessary to test all the possible pathways.  This includes not only correct inputs, but incorrect inputs, so that error handlers can be verified as well.  This provides several advantages:

• you know how the box handles errors
• you can usually write tests that verify all code pathways
• the unit test, being more complete, is a kind of documentation guideline that the implementer can use when actually writing the code in the box
• resource dependencies are known
• internal workings can be inspected

In the "write the test first" scenario, the ability to write complete tests is vital information to the person that ultimately implements the code, therefore a good white box unit test must ensure that, at least conceptually, all the different pathways are exercised.

Another benefit of white box testing is the ability for the unit test to inspect the internal state of the box after the test has been run.  This can be useful to ensure that internal information is in the correct state, regardless of whether the output was correct.  Even though classes are often implemented with many private methods and accessors.  with C# and reflection, unit tests can be written which provide you the ability to invoke private methods and set/inspect private properties.