As I was unsatisfied with the answers already given, I decided to explore the data that is readily available from the Senate website. From the Senate votes website, it is possible to get an xml spreadsheet for past roll call votes for each US Congress, starting from the 101st Congress.
To answer the question of why the Vice President would cast a negative vote, we must first develop a model of how the Vice President would vote in a given situation.
From the 101st Congress to the present, the Vice President has cast a vote a total 48 times. In each of these votes, the Vice President voted in the affirmative. However, in that same period of time, there have been a total of 104 votes that were equally divided and required simple majority vote to pass. As a side note, the Constitution does grant the Vice President a vote whenever the Senate is equally divided and, strictly speaking, there have been 145 instances of this since the 101st Congress, but 41 of these votes involved Questions that require either 3/5ths approval or 2/3rds approval. For example, the vote was split 50-50 on whether or not to convict Bill Clinton of Article Two in the Articles of Impeachment against Clinton and hypothetically a casting vote could have been cast here without any effect.
In the 48 cases where the Vice President has voted in the affirmative, the Vice President's party has always had more Senators vote in the affirmative than in the negative. The only times the Vice President's party has had substantive disagreement (defined as less than 40 Senators from the Vice President's party in support) on the Question for which the vote was tied was Senate vote 255 of the 2nd session of the 103rd Congress (motion to table johnston amdt no. 2446 on H.R. 4624) and Senate vote 119 of the 2nd Session of the 107th Congress (Motion to Table Allen Amdt. No. 3406). In both of these votes less than 40 Senators from the Vice President's party supported the Question at hand.
One can then make the general conclusion that if the Vice President casts a "Yea" vote, then there is at least “majority of the majority” support for that issue within the Vice President's party (at least since the 101st Congress).
Unfortunately, the number of "Nay" votes from the Vice President are limited and all of these "Nay" votes are from prior to the 101st Congress. However, the times the Vice President does not cast a tie breaking vote are quite telling.
Since the 101st Congress, the Vice President could have cast a tie breaking vote 56 more times, but did not. On 16 of these occasions, the majority of the Vice President's party supported the affirmative side of the issue at hand, see the list of votes here; however, the Vice President did not cast a vote in the affirmative. There are a number of reasons this could have happened, but I believe the most likely reasons are:
- The Vice President was not informed, the Vice President was not told that the vote would be close, or leadership did not realize the vote would be tied.
- The Vice President did not want to take a public stand on that issue.
One of example of the first reason could be vote 237 of the 1st session of the 103rd Congress. Here the Question was "to Table (motion to table: is harkin amdt. no. 756, germane)" and on this issue the motion failed. However, it was clear this amendment was not going anywhere because immediately after the vote to table this motion failed, the amendment fell because 54 Senators voted that the amendment was not germane. This seems to be an issue, where either the leadership did not expect the tied vote or the Vice President was not informed because the amendment would die anyway.
One of the examples of the second reason was the Boxer amendment No. 2873 to Senate Bill 1134 in the 2nd session of the 106th Congress. During the time this vote was taken, Al Gore was in the midst of his 2000 Presidential campaign and may have not wanted to be seen as anti-gun. The amendment stated that "the United States Congress has failed to pass reasonable, common-sense gun control measures that would help to make schools safer, improve the learning environment, and stem the tide of gun violence in America" (among other anti-gun language). Here, Al Gore probably did not want to make enemies of the gun lobby or the anti-gun lobby while he was running for President and skipped the vote to maintain "strategic" political ambiguity.
From these case studies, it becomes evident that the Vice President very rarely casts a "Nay" vote.
A "Nay" vote is only cast when the Vice President wants to make it clear for political purposes that the Vice President opposes a particular issue or supports its converse; this wikipedia article provides a list of casting votes of the Vice President since 1945 and we see the few "Nay" votes cast allowed the Vice President to take a clear political stance: Agnew supporting the Safeguard missile program, Bush voting against a motion to reconsider a nomination and showing his support for a conservative judge. Lastly, in the last few decades, the Vice President seems to have abandoned casting performative "Nay" votes altogether.