Vanity pets (such as pigs, monkeys dog, or ANY pet for sale in the crown store) are just that.. vanity. The pets do nothing but follow you around. They do not add any aid in battle or in any other way, they are a purely cosmetic item.
If you are talking about the Twilight or Clannfear you see some players have, those are summoned pets only available by sorcerers, and it's actually a skill that takes up an ability slot.
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Well Mounts are an incredibly useful asset and are pretty essential to have fun in PvP while pets just bug out and sometimes you turn around and they aren't even their so..yeah..I'm fairly certain if given the choice everyone would pick a Mount over a Pet.