The vice president has zero power unless the president is incapacitated or the senate is tied or there is a join session of congress ( and here the role is more or less ceremonial). Anything beyond that is a matter of what the president chooses to delegate to him and his power of persuasiveness. Obviously, some presidents have delegated more than others and some vice presidents are more persuasive. A president could choose to rely heavily on the vp as Bush did with Cheney, or he could never see him or speak to him again after the elections - basically tell him to go home and they will call if they need him.
