Just a quick call to the technical gurus!!!!
My kite board is a quad and I love it, I have about 7 surfboards that are all thusters but my new Xmas present (old man surfboard 6.6 21" wide,huge volume) has 5 slots.
If anyone can enlighten me as the advantages and disadvantages of each set up, I will promptly test the theories
... my reply is not going to be very technical but my experiences might help.
Quad vs Tri made a huge difference for me on my board (Airush Cypher 5'10"). I learned on a directional a million years ago and quickly went to TT in the beginning so going back to directional boards was like learning again, especially strapless. Having 5 fin boxes, I didn't know which way I would like it so I did some experiments. And to muddy the waters, my board's fin boxes have the ability to adjust the position fore/aft by a couple of cm too.
Experiment one was since the board came with 4 fins as standard, I put all of them in and all the way back then got used to the board.
My next experiment was to take rear pair of fins out and try the tri configuration and I instantly liked it. The board tracked better and I felt I could go upwind a little better. Some how, less fin area gave me more drive upwind and it didn't feel so twitchy.
Next I pushed all 3 fins forward in the boxes and it loosened up, still a smooth feel and way more manourverable but without the twitchy feel I felt in 4 fin mode. I get to snap gybes quicker than when fins were back but not throw me off like the 4 fins did. I think since the rear fin can be 'broken' out easier than 2 rear fins, its alot more forgiving and makes the board reward little mistakes instead of kicking you off.
So, from what I found, if your more experienced and aggressive and want more snap, 4 fins is the go, but if your in the mood for more carve, drawn out gybes, tri fin is the go.
Hope this helps?
cheers for now,
Robbie 
Quad is faster down the line and looser/skatey feel. I like quads cause I kite in scrappy small onshore waves and they feel fun! But in larger waves I'd be inclined to aim towards tri fin. Get out and try different configurations, you'll notice a difference straight away.
I think you test it out for yourself and just see what suits your style of riding.
I'm with Doggie I love the fcs knubster fin with the quad, I like the fast and skately feel of the quad, but sometimes the on/off feel during a turn (especially high speed drawn out turns) can feel unsteady compared with a thruster setup. For me the knubster fin smoothes out quad feel and still maintains the speed and quad characteristics with just a little more hold.
Have fun experimenting
From a surfing background, I would qualify these fin options. There should be an optimum set up for each shape, in particular the tail, however that may vary according to personal preference. Person A may prefer a thruster set up because they like the stable drive, whereas person B will go for the quad because they like the turning response.
I prefer a quad for smaller waves and a thruster for head high and above. That said my kite board directional is a quad, but I'm still wrapping my head around it.
My advice is to try a variety of fin set ups in a variety of conditions and find what feels best for you.
