It doesn't run "programs" like a Turing or von Neumann machine does; it's not even a universal quantum computer. It solves certain specific instances of an optimization problem which, in the general case, is NP-hard.
(Also equivalently and more commonly known as the Ising Model). The D-wave chip is just a very fast hardware-based Ising model solver. You can use it to both find the minimum of a problem through cooling, or by controlling the temperature of the system you can also do sampling. And since so many important problems can be reduced to an Ising model, this is interesting.
What I don't fully understand is why is it so important to show that the device takes advantage of Quantum effects functionally? A classical super-efficient Ising Model solver is interesting enough.
Also, the video in the article is really quite terrible. I only half understood it because I know what they were trying to say, but if I didn't I think I'd be very confused at best, and amusingly misinformed at worst.
Just to correct your first statement, in their recent article, they have ruled out idea that D-wave solves problems by cooling (otherwise known as simulated or classical annealing). Quantum annealing, a special case of Adiabatic Quantum Computing, is a completely different approach for solving Ising problems.
Unconstrained optimization is NP-hard problem, and quantum approaches won't change that. However, D-wave hopes they are faster in solving optimizations than classical solvers, something they haven't demonstrated yet. The fact whether D-wave is actually a Quantum machine is interesting to Computer Scientists and Physicist who care about accuracy of commercial hype, and also to D-wave's future customers.
Some prior paper demonstrated that a carefully crafter computer program was as fast as this machine, and I think they did not believe the Q system would get asymptotically faster.
