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Supporters of both methods tend to use 3 units of crisis suits. The Mont'Ka favouring Hammerheads in heavy support over Broadsides the favorite of the Kauyon.
I find hammerheads are essential in taking heat off the more fragile devilfishs which are transporting the fragile Tau troops, to give mobility to railgun shots to allow some side shots on enemy vehicles, and lastly as good objective contesters. Therefore I have always used two, backed up by a unit of broadsides, and I believe that most people start with that set-up.
Given that alot of armies tend to have 3 units of crisis suits in elites, and 2 hammerheads and a broadside unit in heavy support, it seems to be the troops and fast attack slots that makes the tactics of the army differ.
Tau troops are fragile and not that good for their points from a shooting point of view, both types of army will tend to take as few as possible for the objective taking that they think they need. So whether it's a fire warrior unit in a devilfishs or 2 kroot units both methods try to keep their troops cheap as required to capture objectives so they do not take away too many points from the firepower of the army.
If Mont'ka is supposed to sacrifice firepower for mobility, what extra does the Kauyon have that should give it much more firepower. Although the first easy answer is Broadsides instead of Hammerheads I believe the real answer is Pathfinders. Mont'ka doesn't want to add static elements to the army which has to defended or can't move to where the enemy is, while Kauyon loves the extra damage the Pathfinders allows the rest of the army to do.
If you think of the pathfinders without their devilfish (which should be given to a unit of fire warriors), then the extra damage 8 of them will add, for less than 100 points, is remarkable. The change in tactics they demand is that sometimes you will be forced to protect them which will tether at least some of your army to be near their location. This will force you to use kroot as sacrifical /speedbumps units. Not only do you not want the enemy to get to the pathfinders you don't want them to get to any nearby units.
With the most cost effective firepower units left in the Tau codex being mobile (apart from one), ie. Shas'el, Crisis Suits, Hammerheads, the Kauyon way is to increase that firepower more, and to me the best way is mainly Pathfinders (aided by considering switching to second unit of broadsides at the expense of a hamerhead).
The use of Pathfinders and the tethering of some of your army nearby (due to their static nature) will require kroot screens (and if you like them, piranha blockers). Pathfinders require you guess where the enemy will come to ensure the pathfinders have line of sight to contruct the perfect killing zone (Kauyon). Without being restricted by the static area of effect of pathfinders an army can be free-er to hunt down their enemy (Mont'ka).
Rathstar
As a parting note, as army size increases (at 1850 pts and beyond) even crisis suits will run out of space to run and may require kroot screens. Also if LOS blocking terrain is commonplace then the effectiveness of pathfinders and broadsides could be serverly weakened. These two points show that this analysis could be flawed when looking at your local playing environment, but should give some food for thought on the style of army list you want to use.
Late Edit: Although Broadsides also add static elements to the army, they only add better anti-tank, and can normally be allowed to die after the first three turns of the game by when the essential tank killings needs to happen. However pathfinders add to the type of firepower that is needed that turn, eg. against a tank strip it's cover save for broadsides, for a horde guide missile & burst cannon suits, against MEQ guide Fireknifes or Helios in rapid fire range. If you can protect them you can improve the element of your army that is most required turn by turn. Broadsides also don't tether part of your army nearby as much as pathfinders.