Hooke and Newton did have a dispute on the nature of light, but it was not both claiming the same discovery. Hooke argued for light behaving as a wave; Newton said it acted like a particle. Hooke's response was a campaign of slander and defamation against Newton. As an odd side note, both men were right, sort of. Today the wave/particle duality of photons is taught to all physics students.
In Newton's day many folks understood that the force we call gravity must exist, and Hooke had vaguely suggested gravity weakened with distance. But no one could come up with a mathematical approach to accurately describe how gravitational attraction changed over distance. Newton did not discover gravity, but he did precisely define it.