In what fundamental way did the work of Galileo differ from his predecessors who had thought about the sky?

Answer: The correct answer is:

Galileo used instruments and experiments to show him what nature was doing, instead of relying on pure logic

Explanation:

Galileo can be considered as one of the precursors of experimentation and the scientific method. A method that doesn't rely on "common sense" and rationalization and logic, but instead is fuelled by a disposition of skepticism and rather makes claims about reality based on experimentation and empirical data shows.

Galileo differed from his predecessors because he actually used and developed instruments and method to reliable measure and observe what nature was doing, instead of relying on pure logic.