Basically, it mostly matters to the theoretical-physicists-based economy. To all these people for whom the validity of the Relativity is required (together with all what follows to it, like Quantum Mechanics), because in case of being proven wrong (and/or useless), lots of big projects/reputations/money-generations might be lost.
It is true that we don't have a direct application of this at the moment, but most things don't have direct applications when they are initially discovered. Electricity didn't, nor did radioactivity, nor did relativity (which is actually applied in GPS systems). But there are a lot of people who care about this who aren't theoretical physics people (I'm a mathematician for example.) Moreover, there would actually be more likely be money and new big projects if the gravity waves didn't meet the theoretical expectations. It is much easier to get money for research when a field is in a state of complete confusion.
Finally, please note that quantum mechanics doesn't follow relativity, but is essentially orthogonal. One can do quantum mechanics in a completely classical space-time (and in fact that's much easier). Special relativity can be made consistent with QM, but we cannot at this point reconcile quantum mechanics and GR.