It seems to me that the use of "correlation" varies a bit among the "sciences". And the softer the science the greater the reliance on statistical analysis and correlation.

Mathematics has no use for correlation whatever, except in theoretical work which defines and investigates, but does not actually use, the concept. Of course, mathematics is not really a science since it does depend on a relation to the physical world and is not concerned with experimentation.

Physics is fundamentally grounded in "laws" that one at least expects to hold 100% of the time. When the correlation of predictions from those laws deviates from experimental data, outside of error bounds on the experimental data, then one looks for new and better laws. Physicists generally work on the premise that there are precise laws of nature that can in principle be understood. Even the non-deterministic laws of quantum mechanics are viewed as true laws and not just "correlations". There is causality, and it is meaningful. Confusion of correlation with causality is a grievous mistake.

Other sciences are sometimes satisfied, and are forced to be so by the complexity of the systems with which they deal, with less than perfect prediction and hence use correlation a bit more freely. In those areas the basic principles do not have the same weight as "laws" do in physics. If one were to be sufficiently loose as to call economics a science, then one would be hard pressed to find a principle with 100% correlation to anything. Biology in many areas must also accept less than perfect correlation.

Reliance on correlation comes with a steep price. Witness the numerous early correlations in medicine that have turned out to be misleading.

The role of correlation vs causation in science in general is probably too general a topic. It is too dependent on the specific discipline.

But mindless data mining without the organization of data via unifying principles is not good science in any discipline. It is not even good scholarship.