Not always. In the movie Zombie Flesh Eaters AKA - Zombie Creeping Flesh, they were depicted as Apex killers that revelled in chasing down, tiring out and killing their victims.
This was depicted brilliantly in the fight between the "Great White" Shark and The zombie.
The Zombie is now established as the most proficient killing machine in the world.

Mummies were however, people of power and wealth in their first life, and a curse upon the person that disturbs their rest will be chased until their demise, destroying everyone and everything that stands in their path.
They are not zombies, as they were dissected, and had no brain to speak of, just mindless automatons carrying out the whims of the curse, a vessel for the curse if you will.
They were animated spirits, not living dead.
They cannot be killed unless the curse is lifted.
Zombies will keep going until the virus is eradicated or they're all killed, and often relearn their human traits, given enough time being reanimated.
The ravenous state, is merely their baby stage.
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Tl;dr: Mummies are BluRay downloads on a blank USB.
Zombies are the original BluRay, but it's been left out of its case, is full of dust and scratches, and only works on that shitty portable BluRay player you got for a dollar, that only works with certain disks.