Hi Guys,
In short, yes fins make a big difference. Fins offer directional stability and control in turns and jump landings and also through choppy water. It is possible to ride without fins but you will need a lot of concave to make it work well. Like the buzz boards of 2003 (finless concave boards of around 80-90cm)
A friend of mine in Townsville made fins out of aluminium flat strip for his composite home made boards about 2mm thick, they worked a treat and cost about $2 for 4.
I've ridden several boards without fins of both convex and concave design. Any board can be ridden without fins, flatter boards are most difficult.
On speed runs in Augusta (glassy flat water and strong wind), I removed all fins to reduce drag and although it was more slippery through the water, the top end was the same as you couldn't maintain the level of control at extreme speeds.
Convex will make the board handle chop better by reducing the slapping. You gain some control at high speed through rough water, but you lose upwind ability as the board makes more leeway ie side slip. Concave aids in tracking and upwind by peeling the water away from the board and digging the heel rail in at a slightly more vertical angle than normal boards.
Flex gives control by allowing the board to conform to the pressure applied. It also offers spring from the tail and takes some of the bone shattering slapping out of the board in chop. You lose upwind ability with a flexy board, but they carve nicer than a hard board.
A very hard board will be faster and will go upwind better (depending upon rocker) but is not fun with big flat landings from kiteloops etc. A board with progressive flex works best because it offers stiffness in the middle and flex in the tips for carving nice turns and controlled spring off the tail in loaded moves.
Rail design is all about how it handles in waves and rough water. A soft (rounded) rail will suck the board down a fraction more and allow water to slide around the edge of the board. What this allows is the board to not catch a rail and has a more forgiving transition from rail to rail, also allows slower more aggressive turns on waves like on your hammer.
A sharp rail has a cleaner release from the water which is better for speed and high speed carves but works lousy in waves where your speed is slower. Sharp rails tend to be less tolerant of mistakes and you are more likely to catch a rail and face plant yourself big time (stings!)
There are as many different types of fins as there are board shapes and all have a different feel and characteristic. Tall fins, short fins, raked, rounded TE, cutaway etc. They also have different hydrodynamic profiles to create lift at low or high speeds, symetrically or assymetrically, draught forwards or aft or sharp or blunt leading edge.
A new set of fins can dramatically change the characteristics of the feel of your board and it is definately a lot cheaper than buying a new board, plus you can keep fins you like and sell off the ones you dont.
Before buying fins, borrow some from a mate and try before you buy. Specialist fin shapers produce better fins than most production board manufacturers and even different fins amongst the range of boards.
Hope this goes some way to helping, it really is the readers digest of answers but to give a complete answer I'd need to type away for half a day and it would really be quite the tome.
Good winds,