DPS: The PvE Hunter pet's raison d'etre.
In PvE, a Hunter has one role: DPS. Yes, he can bring a little bit of extra utility in the form of traps, dispelling, mana drain, pet off-tanking, and Misdirection, and I shall not deny that, but his ultimate role is to make sure the boss dies fast by optimizing his DPS. One aspect of Hunter DPS is their pet, especially if they are Beastmastery specced. So in order to do the best DPS they can, a Hunter must get a pet who turns out the best DPS. Ravagers, Cats, and Raptors fill this role the most readily, for two reasons.
The first reason is their base damage. Each pet family will have innate modifiers as compared to a baseline of their DPS, HP, and AC. Sometimes it will be a positive modifier, sometimes a negative modifier. In the case of the aforementioned Cats, Ravagers, and Raptors, they all have a +10% modifier to their DPS, which means that they will do more base DPS than most other pet families.
The second reason is their skills. Each of these pet families also has a focus dump. Quite a few pure DPS hunter builds will include one of two talents that provide extra focus to a pet: Go for the Throat and Bestial Discipline. With these abilities in play, chances are your pet will eventually have some excess focus to get rid of. When that time comes, what they need is a focus dump: That is, an instant-cast damage ability that has no cooldown (apart from the global cooldown). Raptors and Cats have this in the form of Claw, while Ravagers have Gore. They also all have Dash, which means they can close with the enemy and begin DPSing quickly. All of this means that they will always be able to use their focus to churn out as much damage as possible, thus contributing to their master's main role in a raid or group.
Cleanse and Abolish Poison: Why Scorpids are the kings of PvP pets.
In PvP, a Hunter is still considered a DPSer, but the fact of the matter is that PvP opponents are bit more wily and have better self-preservation instincts than your average PvE mob. They won't just stand there and get tanked while you plink them with your shot rotation. Thus, in this case, you need to use your utility a bit more often. Scatter Shot and traps play a large role in PvP for a Hunter for sure, but Viper Sting may be one of the Hunter's most infamous and useful Arena abilities.
The problem with Viper Sting is that it can be dispelled rather easily by both Paladins and Druids, taking away a major part of a Hunter's ability to counter them in the Arenas. Improved Stings is one way to deal with this, but it doesn't always do the trick. The other way to deal with this is by stacking additional poisons. While you can do this with Snake Trap, there is the problem of luring the Paladin or Druid into the trap in the first place, and the fact that you may need to save your Trap cooldown for Freezing or Frost Traps. Luckily, there is one other option in this case: the Scorpid pet.
Scorpid Poison is the secret to the success of many an Arena hunter. It is a Poison-based DoT that stacks up to five times. What this means is that if you let your Scorpid stack it on your target and then let off a Viper Sting, Your Paladin will have to Cleanse the target up to 6 times (possibly even more with Improved Stings) to get the Viper sting off. Likewise, a Druid's Abolish Poison will have to tick up to 6 times (that's more than one cast worth of ticks) to rid it of Viper Sting. It's still a bit of a roulette, but there is a strong chance that the Viper Sting will have stayed up a lot longer than it would have had your pet not been a Scorpid.
The Final Verdict: Does this mean you absolutely need to use these families?
It's true that these pet families will probably be the kings of Hunter pets unless Blizzard starts looking seriously at adding new pet skills again, and they will probably all feel omnipresent for quite some time to come. There's little arguing that they aren't the absolute best for their roles. Still, there are Hunters who cling tenaciously to some of the other "inferior" pet families. I should know, I'm one of them. Are these Hunters crazy? Can you really get along with one of those other pet families? I say you can.
While some other families might not do quite as much DPS in PvE, they can have some utility that still makes them useful. The flying pet families have the Screech skill, which is an excellent aggro grabber and a nice small debuff for group situations. In addition, the Bat and Owl families have a 7% DPS boost, making them nearly as good as the big three families for PvE DPS. Boars are hurting a bit since the Charge/Growl nerf, but they still have the Gore focus dump, and Charge provides a brutal opening hit. Finally, there's been some argument among certain Hunters that if you can get high enough crit to be constantly supplying your pet with focus via Go for the Throat, a Wind Serpent's Lightning Breath will provide the highest pet DPS possible.
In PvP, you may have already realized that Scorpids are technically highly situational. If the player or team you are fighting against has no way to remove poisons, you would technically be better off with a pet that can do a bit more DPS, or that has a stamina bonus to take a few more hits. Also, in Battlegrounds, the fighting is chaotic enough that there will probably be enough debuffs on the target to keep your Viper Sting masked even without Scorpid Poison, if they're even able to get their teammates to cleanse them at all.
Even beyond all that, there is a simple truth that some Hunters aren't concerned as much about what their pet can do as much as how cool it looks. Honestly, I think there's a place for those Hunters too. In the end, if your Raid or group leader or your PvP team members are willing to take a slight hit to DPS or utility, and you still have a solid grasp of things like Pet control and shot rotations, It's my opinion, at least, that you should do well enough, while still fighting alongside your favorite neglected pet family.