Why would you use a non-contact voltage detector if you have a multimeter?

Q2: Non-contact voltage detectors don't need to physically touch the measured hardware. They work by sensing the AC electric field created by live AC wires. They're used to quickly see if there is something live, e.g. connected to a wall outlet, inside a device. You can also use them to track where power wiring is going inside a wall etc. They often also have a metal detector so you can also track wires which are not live.

Q1: "Working on a circuit" doesn't necessarily mean "debugging a circuit" here; you might not have a need to measure the circuit electrically. You might be replacing a broken plastic casing for example, and you'd use the detector to see if there's something that could give you an electric shock at the other side of the plastic before you start to fix it.

Obviously you shouldn't rely just on the meter for your safety. You'd disconnect wall fuses or kill main switches before you start, using the meter just gives you some extra protection: with the meter you might notice for example that you've disconnected the wrong fuse and the unit is still live.

Checking voltage with a volt meter may not always be convenient. For example older light switches had the wires inserted in the back instead of the sides like modern ones. The switch would have to be removed first to use a volt meter. Also if working on a wiring junction in an electrical box. The wires would have to be exposed before a meter could be used.

The meter is called non contact because there is no metal exposed that contacts anything electrified. The tip of the meter is plastic. The reason it needs to be inserted in an outlet is because the contacts of the outlet are recessed. The meter has to be close enough to the contacts to sense the voltage. The meter can also be held up to insulated wires and up against the side of an outlet or a light switch.

For a voltage reading to appear on your voltmeter, you need a complete loop. If one end of a single phase supply had a wire break, you will not be able to detect that the other cable is live. A non contact voltage tester can detect voltage on individual test points.

A voltmeter tells you if there is continuity from the supply to both test points, it does not tell you if a cable is not live.