The methods of agreement and difference are typically combined,
both in everyday reasoning and in science. Mill called this
combination the joint method of agreement and difference.
Why it is valuable to combine the methods of agreement and
difference? The method of agreement, by itself, provides
evidence that a is sufficient for E. Since the effect can occur
in the absence of any other factor, none of the other factors is
necessary, so we have reason to think that a is sufficient.
To tell whether a is necessary, we need to see whether the effect
can occur in its absence--and that is what the method of
difference tells us.
Thus, the methods of agreement and difference have complementary
strengths. If we are trying to show that a is necessary and
sufficient for E, we need to use the methods in combination.