Vampires and zombies are technically part of the same family of supernatural beings. They are both undead, both reproduce by infecting their victims.
There are two primary differences between vampires and zombies. Vampires maintain their higher brain function. They are still conscious beings that are capable of advanced thought. Many possess genius levels of intelligence. Secondly, they are more resilient. They do not decay as zombies do. When injured, they can also regenerate to a degree. So they are far more difficult to kill.
Vampires are incapable of feeding on other undead, because they drain a victim's life force via their blood. They do not gain nutrition through the blood itself. Though vampires cannot die, they must feed in order to remain active. If they do not feed, they enter a hibernation state. Zombies do not possess life force so they cannot be used as food.
We can assume that zombies have the same dietary needs as vampires. They drain life force from their victims to remain active. Walking Dead has shown that they do hibernate if they don't eat.
We know vampires are immune to zombies. Would zombies attack vampires? According to Walking Dead, zombies detect the living based on scent. Vampires do not smell like a rotting corpse. Though they do not smell like the living either. One would assume that they are undetectable.
The main source of contention between these two groups would be competition for food. Vampires would want to reduce the zombie population, as the hordes have insatiable appetites. Since vampires are "invisible" to zombies, and they retain higher brain function, they would be a formidable foe to zombie hordes. They could effectively slaughter zombies using any weapon they wanted to, without any opposition. Naturally vampires would do this to preserve their food supply.
Most of my vampire knowledge comes a little bit from Dracula and a lot from Salem's Lot.