Actually, the Vice President has enormous power.
There are only two people in the USA who are elected by all 50 states (in a "national election") – the President and the Vice President.
This means that the President cannot fire the Vice President. As both were elected by the people, the VP doesn't work for the President. Each can be removed only through the impeachment process. (See Article 2 Section 4 of the US Constitution)
Traditionally, the person occupying this office has been low key and subservient to the President. But that's done out of loyalty and friendship, not by any constitutional or other legal requirement.
A rogue VP could be disastrous for a sitting President.
The VP commands national media attention, can draw huge crowds and can have a major impact on government, politics, culture, the economy, etc. In fact, he/she is in a strong position to challenge the President on any matter, including the office itself. That's power.
The President, of course, also has leverage. He/She can at any time isolate the Vice President, taking everything from him/her, except that which the President cannot take away – the job itself and the two roles required by the Constitution: President of the Senate and second in line of succession.
The President can also drop the VP as a running mate during a re-election campaign.
But while the Vice President is in office, that person can have huge influence and his/her job is as secure as the President's. So it would be wise for a President to keep the Vice President happy.