I'm going to relay an example of two apparently-different types of pattern-matching mathematical ability that apparently don't always come together from my life.

Despite the username and despite currently working on cell biology, I very nearly got a double major in astronomy in college. In high school I absolutely hated and was not good at calculus. Figuring out how to integrate anything more complicated than a basic polynomial would trip me up something fierce. Actually taking lots of astronomy and physics classes in college rescued my esteem for the subject, if not my ability to do it quickly and easily.

Throughout my astronomy and astrophysics classes I would repeatedly find it quite intuitive to figure out exactly what needed to be calculated and create the correct expressions quite fast and then trip up on doing the actual calculus while many other people would do the calculus right but not know what to actually calculate.

An example that sticks out in my mind: on a homework problem we were to estimate the fraction of the excess heat being given off by Saturn that could be accounted for by the fact that its surface is depleted in helium, presumably due to it sinking down to the core and becoming a distinct differentiated layer (spoiler alert: pretty much all of it). I'd come late to a group homework session and everyone was tripped up with how to do it. In seconds it was quite clear to me exactly what assumptions and shapes and expressions to use, and in a minute or so I had a long pseudo-equation that I then filled in with actual expressions for gravitational potential energy and changing densities and whatnot... leaving a huge horrific integral that it took me quite a long time to figure out how to even approach and I had to redo three times while the rest of the group had it solved very fast. This was a repeating pattern. My friends often reported that looking at a complicated integral it was intuitive to them what patterns existed in it and therefore what manipulations to do to make it actually integratable in the same way it was obvious to me what to calculate in the first place.

Any thoughts on the natures of these apparent different forms of mathematical pattern-matching ability?