Gravitational waves would basically provide us with a model for gravity we can understand thus giving us answers to the only fundamental force we don't quite get. It was also mean gravity is also quantised (providing the waves were caused by the particulate equivalent, the graviton) and with information about the graviton that would lead to a quantum theory of gravity which would be very important. With a quantum theory of gravity in hand we could work towards unifying gravity with the other forces.
So in conclusion both are pretty important as they both help with a Grand Unification Theory, the topmost goal of all of Physics.
Again, I wish I could go into more detail but this is as far as my know how goes. Maybe in 6 years after my shot at University I can help out a little more.
This is getting a little advanced for me but according to the theory, the Higgs is responsible for giving stuff mass but also "symmetry breaking" which explains how the electromagnetic and weak nuclear force are different manifestations of the same thing, hence it would better our understanding of how they combine together in the "electroweak" force.
Also, Mass is a weird property and obviously with mass you get stuff like gravity so I wouldn't be surprised if a theory for gravity popped out of the Higgs Mechanism somewhere.