Ykhjuy

In The Lord of the Rings: The Two Towers after becoming Gandalf the White, Gandalf cast Saruman from the Order and the Council. He also got the power to break Saruman's staff, which was very powerful.

So when Saruman captured Gandalf the Grey on Orthanc, why didn't Saruman do the same? He meant to find the One Ring's location from Gandalf. He could have broken Gandalf's staff, thereby decreasing Gandalf's power.

A wizard's staff is symbolic rather than a source of actual power, and so breaking a wizard's staff has no effect on the wizard's power.

How do we know this? Because Gandalf was able to defeat the Balrog without his staff.

At that moment Gandalf lifted his staff, and crying aloud he smote the bridge before him. The staff broke asunder and fell from his hand.

This was before his battle with the Balrog.

Saruman therefore didn't break Gandalf's staff because there was no need for him to do so and also because doing so would not have accomplished anything.

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.