For starters, vampires can only be killed by sunlight, a wooden stake through the heart or a silver bullet. The bitten vampire , therefore, cannot die and become a zombie. Drinking human blood would help a vampire heal from zombie bites.
A zombie is an animated corpse. The vampire’s bite, consequently, cannot infect dead tissue turning the zombie into a vampire.
The biggest advantage to being a vampire is you can’t die from a drug overdose or alcohol poisoning. Party time! (see AMC’s “The Preacher” tv series.)