The difference is Quad is a 2 card solution and 4 way is a 4 card solution.

Quad SLI is utilized by cards that have 2 GPUs onboard, like the 7900/7950GX2, 9800GX2, and the GTX295. 

4 way SLI is for 4 individual cards, like the 285 classified edition, or the new 470/480 cards.

As far as the boards go,Quad SLI will work in almost any standard SLI board.  4 way SLI requires a board that has 4 different PCIE 16 slots with propper spacing to allow the full size cards as well as have at least 8x or preferably 16x throughput.  As well as natively support SLI of that level, like the X58 Classified 4 way, or the SR2.