The Job of the Vice President
That does not describe current practice. The VP has become, to differing degrees in different administrations, a sort of assistant President. In some cases that means primarily ceremonial duties—Hubert Humphrey under Johnson, as immortalize by Tom Lehrer in "Whatever Became of Hubert?" In other cases, most notably Cheyney under Bush, the VP is sufficiently active to be viewed by critics as the puppetmaster controlling the President.
In part this reflects the striking difference between the original system for electing a VP and the current system. Initially the VP was simply the presidential candidate who came in second. Each elector got to vote for two candidates, so on a strict party-line vote the President could choose his Vice President by telling the electors who supported him whom to cast their second ballot for. But if some electors cast their second vote for their second choice rather than for the candidate endorsed by their first choice, the most popular candidate would end up as President and the second most popular might well end up as Vice President. In such a situation, there would be no reason to expect the President to view the Vice President as someone he wanted on his team.
All of which raises the interesting question of what things would be like if the Twelfth Amendment, which set up the modern system, had never been passed. There would probably still be parties and a party could still run two candidates. But the final outcome might pair up the two presidential candidates as President and Vice President, it might make make the "Vice Presidential candidate" (insofar as there was one--the candidate who the party ranked second) President and the Presidential candidate VP, it might pair up a presidential candidate of one party with the VP of the other. Barack Obama and Sarah Palin, for instance, would make an interesting administration.
One disadvantage of that system would be the increased risk of assassination if the President and Vice President were from different parties. One advantage, arguably, would be increased fairness. In a close election under our system, half the voters get the candidate they want, half get nothing. Under the old system, half get ninety percent of an administration, half ten percent—assuming that, on average, the VP spends ten percent of his term as President.
It would be interesting.