For Stephen Hawking, Nobel comity gives rarely the prize for astronomy, and I think last year was astronomy, so his chances were thin. And I am not familiar with a breakthrough he has made. Maybe his proof that black holes do release some things, a radiation thing. Is that worth a Nobel, I have no idea.
For the application of their discovery (hum rather their experimental breakthrough, as it is more a Nobel in practical thing than in theory, even though that interest mightily quantum mechanic theorists), I'm not sure to be qualified enough to make it understood clearly.
If I may try : principles founding quantum mechanics are extremely difficult to understand mainly because they contradict what we, human, can perceive around us. In quantum mechanics one founding principle is that it is impossible to know both the velocity and the position of a particle at the same time.
If you understand that, you see that it is very difficult to have experience with direct 'proof' of quantum mechanics, as there is always 'uncertainties' in the measurement.
What the Nobel winners have been able to do is to 'trap' singular particular (whether protons or ions) in a near vacuum, which leads to a much easier way to study these particles.
One major invention in computer science is the advance of quantum programming theory and quantum designed computer. The experiments models worthy of the Nobel are a step towards a physical building of such a computer.
You may know that a key element of the upgrade of computers is the size of the components. With quantum computer, the 'bit', or elementary piece of information, could potentially be as little as an electron, whereas nowadays it is more a transistor, and little as they may be now, they are order of magnitude larger than particles. That could lead (I think, as I have never research the subject in details) to computer thousands or millions more powerful and quick.
I think there are others applications too, but I'm even less familiar with them, so I won't speak about them
Hope I was clear, didn't make mistakes and don't appear as a fool
