There's two main kinds of poison/venom/toxin: vasoconstrictors and neurotoxins. The latter is probably what you want, as it works on the nervous system. Vasoconstrictors make your blood vessels, well, constrict. It's the same thing epinephrine does (which is why you always want to inject in the thigh and not lower on a limb). The vessels in your digits are smaller and can "die" quicker if blood supply is cut off, which is where snakes or spiders are probably going to bite you. Your nervous system starts at the brain, goes down the spinal chord and then branches out to go wherever it needs to go. A bite to the neck would be closer to the brain/spine. When a nerve is damaged or blocked everything "downstream" is affected, which is why a broken neck can cause paralysis from the neck down.

Biting someone on the wrist probably won't do too much, it would just interfere with the hand/wrist. But also your story is about vampires so I'm sure you can fudge some details.