Author's Note: Before I launch into the details, I would first like to point out that, technically, everything that follows can be thrown out and discarded. People started using the terms maze and labyrinth interchangeably, so so society adjusted to accommodate this, and thus it is perfectly all right, by society's rule book, to use the two terms interchangeably. But the thing is, people are idiots. There used to be a difference, and there still should be. This article is aimed towards enlightening the people of Strolen's Citadel, and banishing their ignorance. And maybe adding another bit of flair to their gaming worlds.
First, a summary of what the two things are. After, we can get into applicability.
The Maze: The maze is what people generally talk about when they discuss the labyrinth. It is multicursal, which is a fancy way of saying that it has twists and turns, and, in general, tries to trick you, and hide the exit or destination from the person walking it. There are multiple paths you can take, most of which are wrong.
The Labyrinth: The labyrinth is unicursal, which is the thing which so differentiates the maze and labyrinth. It is unicursal, so instead of having multiple paths, it has only one. The path way twist and turn, but you never actually have to chose between going left and going right. The path turns in on itself, and the only way of not reaching the center is by turning around. What is following is two pictures, which show two different types of labyrinths.
The one on the topis called the Chartres labyrinth. It was found on the floor of a cathedral in Chartres, France. The on the bottom is called the classical labyrinth, sometimes the Cretan labyrinth, because it stems from Crete in the classic Greek times. The point of the labyrinth is spiritual in nature. By walking it, it renewed the body-mind-spirit connection. It is a walking meditation.
The Labyrinth: The only exception to the unicursal labyrinth is the Labyrinth. This Labyrinth, referred to with the capital L, is the one in Greek mythology, where King Minos traps the Minotaur (either the son of his wife and a bull, or the son of his wife and Zeus in bull-form), and sends in seven young men and seven young women to feed it. Theseus goes in with the help of Ariadne, promises to marry her, and leaves her on the island of Naxos when he sails away after killing the Minotaur.
Use of the Labyrinth in the Game: The obvious use is a bit of window-dressing. Using a design on the floor of a temple, putting it in a King's throne room, etc., to spice up the game. Another use would be to use it as a quest. Maybe some demon possesses anyone who walks this labyrinth, and the PCs have to combat it. Or it could be used to heal you. If you walk the labyrinth and are humble and respectful to the gods, your wounds close, and your mind-body-spirit connection is renewed.
Use of the Maze in the Game: The maze is very easy to insert. Make some king turn the entrance hall into a maze, so that the PCs have to navigate it to see the king (or make the king have died, and the castle infested with undead, which patrol the maze). Make no one worthy enough to see the king if they fail to collect the five beads which are in the maze just outside. A good maze (like the kind that frustrates the PCs when they can't get to the exit and have walked past this section five times) can be a bad maze on paper. The PCs won't see it, and cannot see the overall pattern, or get a good idea of where to go.