Aairline control includes maintenance, crewing, or overbooking. Non-airline control includes weather or airport-wide issues.
My expertise is specifically focused on New Zealand aviation rights. You may want to check the specific regulations of the country you are flying from or to (e.g., EU261 for Europe).
Within New Zealand, the Consumer Guarantees Act 2003 applies. Since this was within the airline's control, you are entitled to:
Further, you are entitled to reasonable compensation for additional loss suffered (taxis, hotels, meals) up to 10 times the cost of the flight.
Pro Tip: If the airline refuses, call them out. Work with them initially, but if they fail to pay, take them to the Disputes Tribunal.
For international services, the Consumer Guarantees Act does NOT apply. Instead, you are protected by the Civil Aviation Act and the Montreal Convention. Enforcing this is challenging as most international airlines servicing New Zealand don't fully appreciate (or ignore) the Montreal Convention hoping passengers aren't clued up on it. Also note that getting a decision at the tribunal is one thing, but enforcement is another, as a foreign airline may no have any financial assets in New Zealand that the court can seize if they don't pay up and you follow through all the way.
This is trickier: you need to prove the airline didn't take reasonable steps to minimize your losses. However, if they cancel/delay on the day (especially over 8 hours), you are generally entitled to:
Note: If the airline offers an alternative days in advance, they may argue they mitigated their losses. However, if the alternative doesn't suit you, you can argue this in the tribunal too.
Since the airline couldn't control the issue (e.g. weather), they are not legally liable for compensation under New Zealand law.
However, they must still provide a remedy — keep talking to them about your rebooking options to get to your destination.