The Staff can represent a lot of things, including status, positions of office, leadership. What is it? It could be seen as a simple pointing device, as in the conductor's baton, or it can also be used as a weapon. The staff can also be used as a simple walking stick, as Gandalf himself cleverly claims it to be at Meduseld.
I think what Tolkien is playing with is that the Wizard's staff is both a form of wand and a symbol of his position. Of course, a staff is much more
visually impressive than a wand.
But it does seem to perform the same basic function as a wand, and that is to channel something. Wands could be seen to be an invention of modern Wiccans, but they exist in the tarot as one of the suits, and are sometimes also known as
Staves, and are often also depicted in that way.
The staff of a wizard is clearly important in Middle-earth or else Gandalf would not have broken Saruman's. But I think it also serves an additional purpose to channeling 'magic', the staff is also a symbol of the Wizard's position. Gandalf is head of the order when he breaks Saruman's staff, so I think he was also performing the function of casting him out when he broke it.