A correlation is exactly what it sounds like: a co-relation, or relationship — like the correlation between early birds waking up and the sun rising. But corollary is more like a consequence, like the corollary of the rooster crowing because you smacked it in the beak. Both words love the math lab but can hang with the rest of us, too.
A correlation is a relationship, but not a cause and effect one. In statistics, a correlation is some connection between random variables or data values. Since correlation is a kind of relation, it's often followed by "between." Here are some examples of correlations out in the world:
Correlation is not necessarily causation, so I'm sure this has nothing to do with the departure of Rahm Emanuel. (New York Times)
Employment looks good on a resume — There’s a strong correlation between teenagers with part-time jobs and good GPAs and college expectations. (Time)
Corollary, on the other hand, is one thing naturally following another. In mathematics, a theorem is a statement proven true through reasoning. Its corollary is a statement so closely related that it doesn't need to be proven independently. For the rest of us, it's more like the aftermath of something. Let's look at some examples:
The corollary to this finding is that dairy products have no effect on cough, he said. (New York Times)
There is a natural foreign policy corollary: those who see America as a work in progress are less likely to view it as exceptional. (Time)
How can non-math geniuses keep the two straight? Remember that correlation has an equal, though not necessarily defined, relationship. Corollary is more like a consequence.