Yes, both are undead and do feast on humans. But, they are no more in the same family as a dog and cat who both love to mooch off humans and sleep all day long.
Zombies
Don't need sleep
Don't have sexual urges
Have no issues with garlic, crosses or sunlight
Like blood, but also like flesh
Are not intelligent
Are not particularly physically graceful
No language
Vampires:
Intelligent
Charming
Sexual
Crafty
Blood drinkers
Adept Don't like garlic, sunlight or religious icons
Can mesmerize
Have to be invited into a domicile
Amulets can be used against them
Do not show up in mirrors
It would seem that basically vampires have more rules!
What they have in common:
Driven by hunger
Can turn others into their kind
Undead