Focusing on the "neural network" part might be confusing you.
Classification/supervised learning is essentially about learning labels. We have some examples that have already been labeled: cats vs. dogs, suspicious transactions vs legitimate ones, As vs Bs vs...Zs. From those examples, we want to learn some way to assign new, unlabeled instances, to one of those classes.
Reinforcement learning, in contrast, is fundamentally about learning how to behave. Agents learn by interacting with their environment: some combinations of states and actions eventually lead to a reward (which the agent "likes") and others do not. The reward might even be disconnected from the most recent state or action and instead depend on decisions made earlier. The goal is to learn a "policy" that describes what should be done in each state and balances learning more about the environment ("exploration", which may pay off by letting us collect more rewards later) and using what we know about the environment to maximize our current reward intake (exploitation). Games are a particularly good test-bed for reinforcement learning because they have fairly clear states (I have these cards in my hand, or that many lives, etc), actions ("hit me!", "Jump up") and rewards (winnings, scores, levels completed). There's also an obvious parallel with animal behavior, which is where the name originated.
In both cases, neural networks are useful because they are universal function approximators. There's presumably some very complex function that maps data onto labels (e.g., pixels onto {"DOG", "CAT"}) for supervised learning, and states onto action sequences for reinforcement learning. However, we usually don't know what that is, and can't fit it directly, so we let neural networks learn it instead. However, you can do both supervised learning and reinforcement learning without them (in fact, until recently, nearly everyone did).
However, the network typically doesn't get "rewritten" on the fly. Instead, it does something like estimate the value of a state or state-action pair.