This matters quite a bit, because gravity is at the root of a lot of unsolved problems of modern physics.
The observed phenomena called dark matter and dark energy both flow from the differences between what we would expect to happen as a result of gravity with what we can see, and what we atually see through telescopes (and similar instruments).
The two undiscovered particles of the standard model of particle physics are the graviton (a hypothetical equivalent of the photon for gravity), and the Higgs particle, which governs mass and inertia. Particle physcists are hard at work on this problem.
Gravity waves are hard to detect because their effect is subtle:
To detect a general background of gravitational waves, astronomers would need to monitor 20 millisecond pulsars for five to 10 years, with the arrival time of the radio waves determined to an accuracy of 100 nanoseconds, Jenet estimates. Recording gravitational waves from a pair of merging supermassive black holes would require five pulsars with radio wave arrival times known to an even higher accuracy, 10 nanoseconds, he adds.
“We currently have about 20 millisecond pulsars, but only five of these can be timed to the needed precision[.]”
Several big budget efforts to observe gravitational waves have thus far failed to see them, despite the fact that the theory necessary to predict them and quantify their magnitude has been in place for 94 years, and has been confirmed in a wide variety of other contexts.
In contrast, the speed of light was known with precision from direct measurement prior to 1850. Phenomena that either produced the right number, but lacked a well established theory to tie it to the speed of light, was in place more than a century earlier. A theoretical basis for determining the speed of light from unrelated easier to observe constants, and many of light's other properties was in place by 1864 (Maxwell's Equations).
We understand the weak nuclear force, which was discovered much later than general relativity and electro-magnetism, much better than we do gravity, although the fourth fundamental force of nature, the strong nuclear force is, like gravitational waves difficult to observe directly.
Gravitational waves are less controversial than many of the unsolved problems of modern physics, however, because all serious scientific explanations of gravity predict that gravity will have waves at more or less the same order of magnitude. This is because any mathematical model that reproduces the conclusions of general relativity that have been confirmed by experiment must have this feature.
In addition to confirming existing accepted theories, the detection of gravitational waves and experimental confirmation of the speed at which they propogate would be an important data point to understand the properties of the cosmos and particle physics. Light propogates at different speeds in different media. Gravitational waves are believed to have the same speed of light speed in a vacuum, but the impact of different media on this propogation is not experimentally established. They could also serve as a medium similar to different frequencies of light waves, with which to observe the early universe telescopically.