Pets have different abilities and specializations.
Your stats don’t change depending on your pet.
I’ll make a basic guide for you below here. 
Pets come in 3 specialization categories which are Tenacity, Ferocity, and Cunning. Pets are VERY situational depending on your content, especially in PvP. But here are the basics you need to know:
Tenacity:
- Gives you 5% increased maximum health when summoned.
- Gives you the Fortitude of the Bear spell (20% healing and increased health for 10sec).
Essentially you wanna use Tenacity pets when you are doing solo content that require extra healing and tankyness, and this one is my go-to type pet for solo play.
Fortitude of the Bear is also a great spell to press right before Exhilaration, since it increases how much Exhilaration heals you.
Ferocity:
- Gives you 4% leech.
- Gives you the Primal Rage spell, which is Bloodlust/Heroism.
This pet type got nerfed by a lot, since the leech used to be 10%…
Essentially it is mainly just a pet you use to get Bloodlust/Heroism for a dungeon/raid fight, if no other person has Bloodlust/Heroism. So while it DOES heal you, it’s in small constant amounts that you don’t control “when”, like you can with a Tenacity pet.
Cunning:
- Gives you 8% increased movement speed.
- Gives you the Master’s Call spell, which gets rid of roots or slow effects on you for 4 sec.
Cunning pets are VERY popular in PvP, since every class today have a slow, or root. So because Hunters constantly gotta move to not get hit, Master’s Call is a great PvP spell to have to get away. However in any fight against classes that doesn’t rely on roots, it’s often better to use a Tenacity pet for the extra healing/tankyness. Cunning pets are rarely used in PvE from my understanding.
Pet Families:
Pet families are different from “pet specializations” (Tenacity, Ferocity, Cunning), since every family have extra abilities that you get on top of the above mentioned pet specialization bonuses.
These range from your pet being able to slow your enemy, to decrease the healing they receive, to reduce damage the pet takes… etc…
The most popular ones for PvP, are pets with the Mortal Wounds or Slow abilities. For PvE I don’t know, but I think your common sense can pick one that suits your need. 
Here is a list of the abilities for each specialization: hhttps://www.wow-petopia.com/abilities.php (remove the “h” in front of the link)
Stat priority:
Your basic stat priority for Beast Master Hunter (the one with 2 pets), are the following:
PvP (Dark Ranger hero talent):
- Agility;
- Versatility to at least 20%;
- Haste to 20%;
- Mastery/Versatility;
- Versatility over 25%;
- Haste over 22%;
- Critical Strike.
PvE (Pack Leader hero talent):
- Critical Strike;
- Haste;
- Mastery;
- Versatility.
PvE: (Dark Ranger hero talent):
- Haste;
- Critical Strike;
- Mastery;
- Versatility.
Pet visual changes in patch 11.1:
In regards to “chosing your pet based on looks, and not specialization”, we are getting updated pet changes in the next patch which are AMAZING! 
Essentially you can just pick and choose which specialization your pet should have, when you visit the Stable Master.
This only counts for specializations though… And not pet abilities…
So if you want a “Devilsaur” (dino), which currently has the Ferocity specialization, then you can change it into the Tenacity specialization instead if that is your wish.
However… You can’t change its ability, which is currently Mortal Wounds…
(but this is probably more impactful in PvP)
So if you PvE and don’t care about pet abilities as much, then you will basically be able to pick ANY pet you want, and just change it’s specialization at the Stable Master.
