I understand the concept of tunneling only from a very basic particle encountering a barrier point of view. If I have a finite square barrier of energy height $D$, I expect classically no probability of finding the particle of energy $E<D$ to be able to pass through. This doesn't apply for the quantum case - and this is what we call tunneling.
Now, we move on to annealing. Simulated annealing considers a potential landscape where one starts at a random point in the landscape and tries to find the global minimum. To avoid being stuck at local minima, the algorithm allows you to jump to a higher energy neighbour with some probability. However, for local minima that are very deep and narrow, this is not enough.
Quantum annealing represents the potential landscape using an Ising model for a bunch of spins in a magnetic field. There is a temperature parameter that allows the system to move to states of higher energy, analogous to simulated annealing. A claim is often made that for deep and narrow local minima, the quantum state "tunnels" out, where classical simulated annealing would stay stuck. The image below is from Wikipedia and shows the thermal jump of simulated annealing vs the tunneling in quantum annealing.
In what sense is this tunneling related to the particle and barrier scenario? Is there a better/different way that I should think about tunneling e.g. in terms of the state of a system being able to overcome a classically impossible barrier?