Vrftjkry

Presumably because Gandalf was sent back by the Powers with greater power specifically for the purpose of fulfilling the role that Saruman had forsaken, to contest Sauron.

From The Two Towers (emphasis mine):

Naked I was sent back - for a brief time, until my task is done.

Then to Saruman later:

'Come back, Saruman!' Gandalf said in a commanding voice. To the amazement of the others, Saruman turned back as if dragged against his will...

Since the Istari were Maiar, an increase in power would only have come from some greater being (i.e. the Illuvatar) so Gandalf is able to do what he did to Saruman (a reversal of their earlier position) because he is an agent of a greater Power still, and was granted that authority by his divine master.

I always saw breaking the staff as a sign of supremacy.

Saruman already outranked Gandalf when he defeated him. Breaking Gandalf's staff wouldn't serve any additional purpose; the higher-ranking wizard should have been expected to win anyway.

When Gandalf later defeats Saruman, Gandalf breaks the staff as a symbol that his power has become greater, and he assumes the dominant rank.

As to the staffs themselves... I think a wizard does gain benefit from their staff, but it's not because of power in the staff itself. It's merely the benefit you get from a tool that has become comfortable with long use, but still one you could make again if need be.

Saruman didn't break the staff, because he managed to take it whole from Gandalf and kept it (check out the end of the fight where Saruman has both staffs).

When Gandalf flees Orthanc he does so without his staff (revisit the scene here) and later on gets another staff. This differs in the films from the books. The accounts of how he gets another staff in the movie universe differ a bit and maybe this is worth a separate question, but he clearly loses his staff in the fight with Saruman.

Somewhat authoritative sources for this answer are this IMDB FAQ (check out How did Gandalf get his staff back from Saruman? He simply leaps off of Orthanc without it) and the Lotr wiki.

The flight scene also shows that Saruman still hoped he could turn Gandalf until that moment, so it was reasonable to keep his staff intact, but out of reach of Gandalf. Also note that in the movie universe, wizards obviously can make use of the staffs of other wizards (perhaps less efficiently). So, even if he would not convince Gandalf it might be prudent to keep the staff just like you might keep a spare sword around, if you have it anyway.

Poetry and drama. Gandalf's line, "Saruman, your staff is broken," is the only quote I remember from The Two Towers. It is not a direct command, it is as just as much a comment on how the positions have changed between these two wizards as well as destructive magic.

The storyline now apparently finishes with Saruman and the struggle for Middle-earth now turns towards Mordor.

It might be that Saruman, possessing the staff of Gandalf, could have more control over him. While Gandalf broke his staff since he did not care about having control over Saruman.

It appears, from the Lord of the Rings books, that a wizard's staff is a method of channelling their respective energies.

For instance, in The Fellowship of the Ring, Book 1:

..."The figure lifted his arms and a light flashed from the staff that he wielded. A mighty eagle swept down and bore him away."...

In Book 2:

..."...he thrust the end of his staff into the midst of it. At once a great spout of green and blue flame sprang out,..."

There are also many mentions of Gandalf lighting the way through Moria using his staff, and guiding the party (using the staff as a beacon).

As pointed out by Victim of Circumstance, Gandalf strikes the bridge with his power, most likely channelled through his staff to a point of force at the staff's end, so much power that it destroyed the staff. user1643723 is absolutely right in his comment - there is a power there, but maybe not within the staff itself.

Later, in The Two Towers, Book 3, we find Gandalf in Edoras. Háma states:

"The staff in the hand of a wizard may be more than just a prop for age."

There are many more references to the power of the staff throughout the books (a Kindle is a great resource!).

So, it could be that the wizard's staff is more than just an emblem of power - it could be a conduit to channel such power to a fine point, or a focus to bring that power to bear.