Quantum computations are a case of reversible computations. If you're aiming to relate qantum computers to other kinds of computers, I suggest starting with the paper called "Synthesis and Optimization of Reversible Circuits - A Survey" [0]
It describes the kinds of computations that need reversible computations and kinds of devices that can perform them. That makes it easier to notice similarities, as quantum computers are not the only ones in this category. Notably, photonic and low power computations call for reversibility.
The first couple of pages of the paper also touch upon what can be done with them - while you don't have to throw away all you knew about logic gates, the reversible ones are their own thing.
If the water goes too deep, perhaps Mika Hirvensalo's "Quantum computing" helps. It approaches the topics of how to compute with qubits from mathematical side.
I still don't know the theory behind quantum annealing though - I don't think these two cover it.
Still, I learned that the main difference is that the state doesn't have to travel as with electronic circuits, but rather one would likely "apply" gates as "events" acting on the state. A qubit could then be any quantum object - a electron trapped in one place and being hit with photons for example. Place a couple together and hit them in a coordinated way and that event was your gate.
When it comes to saving states, it will be problematic. Of course, it's possible to "save" a qubit to another qubit - with some workarounds for the "no-cloning" principle.
But if you mean saving them to a classical system, it seems it's not possible at all. A qubit is a representation of a probability, and once read it's destroyed. You could try recalculating it several times and get an idea of the probability it represents. However, qubit readouts are not guaranteed to be independent from one another, giving another bit of headache here.
Disclaimer: I'm not a physicist - just a software engineer trying to understand the hype.
[0] https://arxiv.org/abs/1110.2574