They all attack after your turn and have slightly different stats, some having higher damage, but their true worth is in their different gem slots.
Every pet has 3 different gem slots (you start with a single unlocked and when you level your pet, up to ten, you unlock the other two). These slots differ from pet to pet, and determine which gems you can equip on them.
I'm planning for gem farming as the endgame content, with the arena acting as a boss rush and an easier way to obtain them. With gems you will be able to increase your power even further, with percentage based increases.