When would I use Venom Hydra over Mammoth Hydra, and vice versa?

Venom Hydra is more advantageous in almost every situation.

While the pools of acid do less damage per second, they will stack on top of each other, and with 3 heads as opposed to 1, this can mean that it's actually doing 54% damage per second in one location (which can hit multiple mobs), or 18% in 3 different locations.

Venom Hydra also has speed in terms of the fact that Mammoth Hydra's flaming ground is fairly slow to reach it's target.

Venom Hydra is the clear cut winner in situations where an enemy will stand still for extended periods of time (like bosses or if you're grouping with a melee character), as the dps output on targets standing in pools is just insane.

The only time that Mammoth Hydra seems to win out is if lots of enemies are queuing up (say coming through a doorway), but even then, Venom Hydra can hit the first enemy, and when it dies, the second enemy will take it's place which in turn will mean it's now standing on the venom pools. So Mammoth Hydra is only barely better in this situation, while Venom Hydra is far better in all other situations.

While I agree that Venom Hydra is very good. Mammoth Hydra if cast multiple times after it sends out its fire river, will also stack attacks in the same way that Venom Hydra does except with more damage and AoE.

The most commonly used hydras from what I read, are Venom or Arcane Hydra.

Venom Hydra: I personally use this with Blizzard and find it a good combo. As Mr Smooth pointed out, the venom pools add up and do more damage the longer an enemy is standing there - this makes a huge difference, esp. if they're frozen/stunned/busy fighting some meatshield etc. Synergizes well with Blizzard as they spend more time in the pools while moving slower or frozen.

Arcane Hydra: Spits out arcane orbs, which do decent AoE damage all at once instead of requiring enemies to stand around like Venom hydra. Synergizes well with Temporal Flux which adds a slow to arcane attacks.

Massive Hydra vs Venom: Besides the stacking effects of the venom pool; prior to recent patches some monsters that dealt elemental damage also suffered less from the same damage type. So you could do 20% less damage to molten or fire chains with Mammoth (or 20% less damage to Plagued with Venom). I haven't tested this however, and at this point its no longer applicable. Another historical point, the LoK bug worked better with 3 heads than 1. Neither of these change the fact that venom simply outperforms massive in most cases, as Mr Smooth explained in detail.

One other note is that Ice Hydra seems to have a much shorter range than the others; if you try using it keep a close watch on whether its actually doing anything - it cannot attack from long distance unlike venom et al.