So this cartoon has been going around my Facebook friends list
First of all, yes, "correlation is not causation" is strictly true. That is, they are not the same thing. If events A and B tend to occur together, this does not mean that A causes B, or that B causes A. There may be a third, unobserved event C that causes both, or the observed correlation may simply be a coincidence. Bear this in mind.
But if you observe the correlation frequently enough to establish significance, you can be reasonably sure (arbitrarily sure, depending on how many times you make the observation) that it's not coincidence. So now you're back to one of three explanations: A causes B, B causes A, or there exists some C that causes both A and B. (Two caveats: whatever the causal relationships are, they may be very indirect, proceeding through events D, E, F, and G; and the word "significance" has a very precise meaning in this context, so check with your local statistician before using it.) An easy way to check for A-causes-B vs. B-causes-A is by looking at temporal relationships. If you are already wearing your seatbelt when you get in a car crash, you are far more likely to survive than if you aren't, but you have to have made the decision to put the seatbelt on before the crash occurs--it's the fact of you wearing your seatbelt that causes you to get through the crash okay, not the fact that you get through the crash okay that causes you to have been wearing your seatbelt. Unfortunately, the temporal relationships aren't always clear, and even if you can rule out B-causes-A on this basis, it still leaves you to choose between A-causes-B and C-causes-(A,B).
An awful lot of what science does is figuring out what C is, or even if it exists at all. This is where mechanistic knowledge of the universe comes into play. Suppose that emergency departments in particular city start seeing a whole bunch of patients with acute-onset fever and diarrhea. Shortly thereafter, ED's in nearby cities start seeing the same thing, and then the same in cities connected by air travel routes. Patient histories reveal that the diarrhea tends to start about six hours after the onset of fever. Does this mean the fever is causing the diarrhea? Probably not, because these days we know enough about the mechanisms of infectious disease to know that there are lots of pathogens that cause fever, then diarrhea. The epidemiologists' and physicians' job is then to figure out what the pathogen is, how it spreads, and hopefully how best to treat it; while they're doing that, the "correlation is not causation" fanatics will be sticking their fingers in their ears and chanting "la la la I can't hear you," and hoping desperately they don't end their days as dehydrated husks lying on a feces-soaked hospital bed.
The point here is that in most cases, correlation is all we can observe. (Some philosophers of science, a la David Hume, would argue that we never observe causation, but I'm willing to accept "cause of death: gunshot wound to head" and similar extreme cases as direct observation of causal relationships.) Not every patient exposed to the pathogen will get infected. Of those who do, not all will show symptoms. Some symptomatic patients will just get the fever, some will just get the diarrhea. Some will get them at the same time, or the diarrhea first. Medical ethics boards tend to frown on doing controlled experiments with infectious diseases on human subjects, so you have to make what inferences you can with the data you have.
Even with all these limitations, correlation--in this case between exposure and symptoms--is still a powerful tool for uncovering the causal relationships. Most of what we know about human health comes from exactly this kind of analysis, and the same is true for the observational sciences generally. Astronomy, geology, paleontology, large chunks of physics and biology