UPDATE: The Cloture vote passed today.
The United States
Senate votes to limit filibusters by adopting the Cloture rule.
Also note that the vote you linked to was a Cloture vote, requiring a 3/5 supermajority to pass.
However, legislation and nominations to the Supreme Court would
still need 60 votes to invoke Cloture, end debate and kill a filibuster.
Even though the vote requirement for Cloture was reduced to 3/5 of the entire Senate(60 votes)
in 1975, in the intervening years, the filibuster has been increasingly used to obstruct legislation.
Easier than it sounds, proponents of the nuclear option say that the Senate's rules only require a majority vote to change them(meaning only 51
votes would be required to change the Cloture threshold from 60 to 51).
Requiring neither speech, ideology or commitment, in today's Senate
if fewer than 60 senators are willing to first vote to stop debate on a bill(Cloture), then one senator,
alone, can just make a procedural motion that stops all voting on the bill.
The result of much compromise between a minority desirous of keeping some method of checking majority tyranny, a majority desperate to make an end run around a small group of Senate doves, and President will to use his bully pulpit,
Rule 22(Cloture) was passed.
Therefore, while it should only take 51 votes(50 if the Vice-President voted to break a tie) to pass a bill or confirm a nominee in the U.S. Senate, in reality, because of the threat of a filibuster, in order to limit further debate,
the majority must first obtain 60 votes for Cloture on anything even remotely controversial.