I'd recommend reading a review paper on disorders of consciousness, which cover the spectrum of minimally conscious states, vegetative states, and coma. Brain death is outside the context of disorders of consciousness, but is often discussed for comparison. Nico Schiff is one expert in the area; I've attached a reference to a review he coauthored at the bottom and I'll draw mostly from that review in this answer.
Schiff and Fins write, with some bolded emphasis by me:
At present, brain death is diagnosed if clinical evidence of the complete loss of brainstem function is present at the bedside and the patient demonstrates a failure of ventilatory drive in the setting of documented hypercapnia (significant elevation of blood partial pressure of carbon dioxide over a chosen threshold) — the ‘apnea test’. Fulfillment of these criteria should invariably associate with an ‘empty skull’: no sign of any metabolic activity or blood flow measured by positron emission tomography, a ground truth linking the biological model to the clinical assessments
The first two statements are most relevant clinically; PET is not commonly used except in research and rare cases of uncertainty. Biologically, brain death is when the brain is...dead. There is no recoverable neural function. The signs are a lack of any brainstem reflexes, including lack of respiratory drive. Additionally, other potential causes of these symptoms (such as pharmacological causes) must be ruled out.
Coma is a state of unconsciousness, but some brain functions are preserved. Again from Schiff and Fins:
Both comatose and vegetative state patients are unresponsive to environmental stimuli, although reflex movements may be present; in neither condition are goal-directed behaviors initiated. Comatose patients lack state variations and typically remain in a closed-eyes state that is unchanging even when presented with the most vigorous stimulation.
Comatose patients do not meet the criteria for brain death described above, but do have a complete inability to respond to stimuli and do not make purposeful movements besides reflexes. Schiff and Fins (and others) compare this state to deep anesthesia. Though your question posits that there should be some distinction between medically-induced and one that is not, there is really no functional difference. The possibility of recovery is tricky to assess, but generally patients who spend a long time in coma have a worse prognosis than those that do.
The clinical approach is different. A brain dead patient is dead. Nothing can be done to recover function; potential for organ donation can be considered, but there is nothing medically to do for the individual. For a comatose patient, it's more complex, and decisions need to be made regarding the level of medical support to provide given expected prognosis in both probability and extent of recovery.
In brain death, the neurons are dead. The lack of electrical activity is not because they are in some quiescent state, it's because they are dead. With coma, it's less clear except that there are signs that some neural function remains. Coma is really a clinical state, not a biological one. It's not always possible to determine the level of irreversible damage in a comatose patient, nor the extent of possible recovery.
Schiff, N. D., & Fins, J. J. (2016). Brain death and disorders of consciousness. Current biology, 26(13), R572-R576.