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.
If you're unsure about anything just fire me questions, I'm getting to the point where I can explain things more simply but sadly in less detail than the likes of Wiki.
When you first get into the stuff it's a nightmare you go onto most websites and it's just a bunch of jargon and it's hard to decipher what's really going on.
You can do a calculation yourself it's not too difficult, perhaps calculate your own personal wavelength.
For the record Planks Constant is a very small number 6.63x10^-34, which is why if you divide by even a momentum of 10 (momentum = mass x velocity) the wavelength is still tiny.
Whereas particles like electrons have a mass of 9.11x10^-31 (or something) so if you plug that in for a given velocity you actually get a reasonable wavelength. As even though "h" (the symbol for Planks constant) is so tiny you're momentum is also very small, so a reasonable wavelength is achieved. One that will go through clear interference and other wavelike properties.
Posted this in the other thread but it was horribly off topic, in response to you things about subatomic particles.
Everything has both a particle and a wavelike property, macroscopic objects just have such extremely small wavelengths that it no longer matters.
By virtue of Wavelength = Planks Constant/Momentum.
Hence if you know the momentum of a car, for example: 1000kg going at 10 meters per second, then the momentum is 10000 Newtons per second.
Take Planks constant over 10000 and you get the equivalent wavelength of said car which is a very small number.
The truth is there is no difference between "particle" and "wave" they are all the same thing.
And by the way IIRC it's the other way around, observing something collapses the wave function which is basically the probability of observing something in a given space. Nothing works on definites in the universe there is only a "probability" that you'll find an electron at a given location for example.