What power does the Vice President really have?
According to the Constitution, the Vice President is the head of the Senate, but doesn't really explain what that means except to say that he doesn't get to vote except to break a tie.
Aside from that and being next in the line of succession, what power does the Vice President actually have? Can he set the Senate's agenda and force calls to order in the Senate the way the Speaker can in the House?
When thinking about political power and influence it is important to distinguish between formal and informal power. While the law explicitly provides some powers, often times the most significant powers are entirely informal (not defined in law).
The Vice Presidency is a position that largely relies on informal power.
Who Runs the Senate?
According to the Senate website, historically the Vice President's major job was presiding over the Senate. Their job included administering the rules of the Senate, casting tie-breaking votes, and overseeing Senate floor meetings. The Vice President's expenses were also funded through legislative appropriation at that time.
This changed in the 1961 when Lyndon B. Johnson moved the Vice President toward an executive-focused role, rather than a legislative one.
Currently the duties of overseeing the Senate are done by the President pro Tempore of the Senate. However, that position is largely viewed as being ceremonial or administrative rather than important, so it is often delegated to junior Senators to help them learn Senate procedure.
So What Does the Vice President Do Now?
As mentioned in Ryathal's answer (and implied in the question) the Vice President doesn't have much Constitutional authority. However, they have a large degree of authority vested in them by the President (see this BusinessInsider article for an interesting discussion). One interesting facet is that because the Vice President is elected, the President can't fire them - unlike other advisors.
Stemming from this position, the Vice President has statutory (but non-Constitutional) roles - such as being a member of the National Security Council.
The Vice President is executive staff, and an advisor without any particular portfolio. So although they aren't vested power by the Constitution, their status as an executive officer, combined with being elected alongside the President, means that informally they are often have significant influence.
+1, Other points: (1) The VP's power depends heavily on how much they are trusted by the President. Cheney and Gore wielded great influence behind the scenes; Quayle and Agnew, not so much. (2) Unlike any other Presidential advisor, the VP can't be fired, only impeached. (3) The VP is usually in a very good position to win the party's nomination to succeed a 2-term President, if they wish to run.
@indigochild VP was added as a permanent member in response to Truman not knowing about the atomic bomb when he took office. It must have been added after 1947. I'll l look for a citation later, but I think I read that at the Heritage Foundation's website. The Wikipedia page just notes "statutory" next to the VP, but doesn't cite a source.
@indigochild Here it is: https://catalog.archives.gov/id/299860. VP was added in 1949
It's worth saying that the Senate makes its own rules, and it has chosen to make rules that give both VP and Pres. pro Tem. very little power compared to House leadership. It could be otherwise.
@sondra.kinsey: That's not terribly stable, because the party in control of the Senate might not control the White House. So as soon as you get a VP of the opposite party to the Senate majority, the Senate agenda would be under the control of the minority party. I don't think the majority would like that very much, so they would probably change the rules to remove that power in fairly short order.