This is a reasonable question. After all, when a car runs out of gas, the engine stops running, but it doesn't break the engine. If you add gas or oxygen, the engine can start up again. This is even true for some cells and some animals.
Why is it different for the brain?
It turns out that this question has been studied, because loss of oxygen is what causes brain damage in stroke. If brain damage can be prevented, the medical outcome from stroke would be much improved.
The reason for hypoxia-induced brain damage is that the electrochemistry of the brain requires active stabilization to prevent feedback-induced ion overloads. When metabolism slows down, the neurons biochemically short-circuit, analogous to how a nuclear reactor self-destructs if the water pumps are turned off.
In terms of biochemistry, the main mechanism is cascading excitatory depolarization, leading to calcium ion overload, which irreversibly damages the cells biochemical pathways.  This neurochemical cascade is called "excitotoxicity."
More specifically, slowing down the ATP-driven ion pumps leads to sodium ion (Na+) buildup and electrical depolarization of the neuron. This causes glutamate release, which excites other neurons, causing more depolarization, and more glutamate release. The cascading depolarization activates NMDA receptors, letting in calcium ions (Ca2+) which accumulate to toxic levels inside the cell. The high calcium causes causes "lethal metabolic derangement" , probably by changing the shapes of the protein enzymes that drive the biochemical pathways that keep the cell running.
These mechanisms can apparently be blocked, and drugs that are glutamate and NMDA antagonists have been found to slow down or stop brain damage due to loss of oxygen, thus providing a potential benefit to stroke victims. 
 Lee JM, Zipfel GJ, Choi DW (1999). The changing landscape of ischaemic brain injury mechanisms. Nature. ( )
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