Being able to influence the choice and development of my pets' stats, skills and abilities.
Being able to upgrade pets with equipment and items too.
Having a variety of different pets to choose from based on the type and style of combat I want to engage in. That may change from battle to battle.
Directly controlling the timing and usage of pet abilities.
Developing a long-term relationship with particular named pets, usually in connection with taming classes e.g. UO Tamer/WoW Hunter.
Controlling multiple pets at once.
Having a mixture during combat of:
Dumb Pets (Short-Term, no control, simple single function) including swarm pets
Temporary (Summoned/Charmed) Pets (Short-Term, owner controlled, no pet development),
Long Term Pets (Long-Term, owner controlled pets that develop).
Having pets which act on other players in support functions for teams/raids.
Supporting my own pets with my own abilities - e.g. healing, short-term in-combat buffs, debuff/CC removal etc.
Getting support from my own pets - e.g. true tanking, healing, CC, damage redirects, buffs etc.