I think the simplest and clearer answer about this question is given from Roy Osherove in his book The art of Unit Testing (page 85)
The easiest way to tell we’re dealing with a stub is to notice that the stub can never fail the test. The asserts the test uses are always againstthe class under test.
On the other hand, the test will use a mock object to verify whether thetest failed or not. [...]
Again, the mock object is the object we use to see if the test failed or not.
Stub and mock are both fakes.
If you are making assertions against the fake it means you are using the fake as a mock, if you are using the fake only to run the test without assertion over it you are using the fake as a stub.