Quick erase basically adds a flag to the disc that says "I am empty - you can overwrite any data on here", but leaves the data fairly intact (though not accessible through normal methods). It is fairly quick to do (!).
A full erase will take longer (dependent on how much data the disc held) and will actually make the disc as close to actually blank as possible. This may take some time!
I generally use quick, though sometimes if a disc has been playing up a full erase will correct errors. If you're worried about data security, smash the disc into lots of little bits rather than trust a full erase.