What is the difference between maze and labyrinth? Well my basic understanding would probably be that a labyrinth is an underground maze and is a word of essentially Greek origin. Whereas maze is an English word which just refers to a complicated network of paths and passages. Either way the aim in a maze is to get lost or find your way through it.
If you want the more technical definitions here is an extract from wiktionary.org:
1. A labyrinth ; a puzzle consisting of a complicated network of paths or passages, the aim of which is to find one’s way.
2. Any confusing multi-part system.
Etymology: Middle English mase, from an aphetic variant of Middle English masen (“to perplex, bewilder”), or perhaps from Old English *mæs “delusion, bewilderment”. Akin to Old English āmasian (“to perplex, confound”), Icelandic masa (“to chatter”).
1 Maze, especially underground or covered
2 Part of the inner ear
Etymology: From Latin labyrinthus, from Ancient Greek λαβύρινθος (labýrinthos) ‘maze’, from an Anatolian language (compare Lydian labrys ‘double-edged axe’ and -inthos typical of Anatolian placenames).