What is the difference between "double dissociation" and simply having control and experimental groups?
In brain damage/lesion studies, a double dissociation gives evidence that function A and B are, to some degree, implemented in different regions of the brain. In general, a double dissociation shows evidence that A and B are independent of each other.

This is a different kind of claim than saying that function A is implemented in brain region X. For this kind of claim, performing an experiment that lesioned (removed) a brain region in some subjects but not others (obviously not an ethical experiment, but let's ignore that) would be a reasonable way to test the claim. 

However, this kind of experiment wouldn't be able to provide evidence that functions A and B are implemented in independent regions. To illustrate why, let's consider the possible outcomes of the experiment:


Functions A and B both remain in tact or Functions A and B are both disrupted. Clearly this outcome doesn't provide any evidence that A and B are implemented in different regions, since both functions are affected equally by the lesion.
Only function A is disrupted, function B remains in tact. The most that this outcome shows is that region X (the lesioned area) is crucial for function A but not B. However, what is doesn't show is that function A is independent of function B. It could be the case that function A can be disrupted without disrupting function B, but that any disruption of function B will disrupt function A.