Enquiry for the purpose of this discussion is defined as Formal Investigation to determine the facts of why things have gone wrong.
Inquest is defined as an investigation of the facts of a situation, particularly one that had an undesired outcome ( literary ). ; An official enquiry in front of magistrate, coroner, or jury into the facts of a case such as a sudden unexpected death.
Both processes have been the norm every time there is such on occurrences or events that criminal investigation may have been initiated and launched on entirely different angle. Or most often after the conclusion of the criminal investigation to further answer the questions that were not brought out or not relevant to the criminal case.
In most cases an Enquiry or Inquest is called not to find blame or fault but to find out how things gone wrong so corrective measures will be figured out, guidelines and preventive measures will be put in place or Laws be passed to avoid the same from ever happening again in the same situation, and situation if possible, be altered to the point that the event will not happen at all... again.
Sometimes, but only rarely, it is called to find out who are responsible for the wrongdoing so punitive action can be pursued from evidence gathered.
How's the process launched and conducted: Either an Inquest or Enquiry is initiated by any authorities directly responsible and under which jurisdiction the investigation be conducted. For local municipal matter, a municipal council can order such, for provincial, the Premier or the Provincial Parliament, for Federal, the PM or the House.
Who composes such Body: For Judicial Enquiry, it is headed by a judge. Before the start of an Enquiry a mandate which defines the power, limitation, and duration (estimated) is handed down by the passing of appropriate bill or law calling for such. Then, the Head, will set up the venues, the conduct, and will issue summonses, and
notices to all legal participants and consider those with legal standing and then set up the starting date and scheduling of witnesses. And the hearing begins. After the conclusion of the hearing, granted it was not called off or suspended for any reason, the Head of the Enquiry will prepare a report as to Enquiry findings and submit it to the Authority who called an Inquest or to the Public at Large. And if mandated to recommend corrective measures, or actions to be taken, then the Inquiry will also submit reports to the effect following the "findings".
For Inquest- usually is undertaken by coroner for cases of sudden and unexpected death as a result of: Police Officer(s) action; Public Safety (water supply, hydro accidents, recreational facilities failures, medical institutions unexplained or misdiagnosed cases, etc). The process of calling it is about the same and the conduct and manner is also similar. But the inquest usually concern of the findings the faults and causes resulting to the tragedy rather than the person or persons involved. But blames and faults as a results of the inquest are often unavoidable. Sometimes, the whole exercise can only point to the negligence or criminal intent of a single party or limited number of person.
Both Processes, done in the context of law, and proper procedures, are two greatest tools in correcting the mistakes and wrongdoings that could sometimes crop in the business of Government and people interacting with each other. Some of these mistakes are innocents, committed without malicious intents, some are criminal negligence, some are downright criminal. Relying on the Present set up of our justice system alone can not cover and see all these activities in our ever complex and evolving society