TL;DR: Because staves can do everything wands can do, plus some other stuff.
Wand vs. Staff: Lingusitics
First off, it is worth noting that Tolkien was a philologist (a scholar of language), and loved archaic terms. He would have known that the word "wand" was historically associated with sticks much larger than Harry Potter wands - in other words, a wand was a staff, and a staff was a wand. The fact that Tolkien sometimes described Gandalf's staff as a "wand" supports the idea that he didn't see much of a difference between the two terms.
wand (wɒnd)
n.
1. a slender stick or rod, esp. one used by a magician or conjurer.
2. a rod or staff carried as an emblem of one's office or authority.
- Random House Kernerman Webster's College Dictionary
The magician Dr. Faustus and his "wand" in woodcuts from 1620
Staves and Magic:
We don't know what the staves in Tolkien's works actually do, in terms of the use of magic.
It is unknown whether Gandalf required his staff to exercise certain powers. At times it appeared to focus or extend his powers, such as when it emanated light. Exactly how much it aided him in the use of magic is unknown, but Gríma Wormtongue tried to forbid Gandalf from bringing it into Edoras, clearly under the impression that without it Gandalf's power would be limited.
- Tolkien Gateway
But when Saruman imprisons Gandalf atop Orthanc, he doesn't take away Gandalf's staff. This would be very difficult to explain, if Saruman had reason to believe that Gandalf was more powerful with his staff than without it. The fact that Saruman let Gandalf keep his staff when he was imprisoned suggests that Saruman knew Gandalf couldn't do anything useful with the staff that he wouldn't have been able to do without the staff.
This issue is hotly debated, but Tolkien scholar Michael Martinez believes that staves are purely symbolic:
It is clear from a few scenes in the story that Gandalf used his staff (such as to light fires, create a light by which to see in Moria, etc.) for “magical” purposes. However, Gandalf also destroyed his staff when he broke the bridge of Khazad-dum and he was still able to engage in a multi-day battle with the Balrog, in which one or both of them unleashed great power against the other.
Gandalf ultimately defeated the Balrog without his staff, so it should be clear to everyone that he did not need his staff in order to use his Maiaric power. Tolkien did not explain the significance of the Istari’s staves in any writings that have been published to date, but they clearly held symbolic value since Gandalf broke Saruman’s staff and Saruman accused Gandalf of seeking to collect the staves of the “Five Wizards”.
In my opinion a wizard’s staff was a symbol of his office — that is, of his role as an emissary from the Valar. Wormtongue’s insistence that Gandalf be forced to leave his staff at Theoden’s door may thus have been a naive attempt to deprive Gandalf of his power because Wormtongue did not understand the true nature of the Istari.
Instances in which Gandalf's staff appears to be used for magic:
Gandalf does appear to use his staff for specific magical purposes - although we don't know what the staff added to his innate powers in any of these situations:
- Breaking the bridge in Moria
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. A blinding sheet of white flame sprang up. The bridge cracked.
- The Fellowship of the Ring
- Lighting a fire on Caradhras:
At last reluctantly Gandalf himself took a hand. Picking up a faggot he held it aloft for a moment, and then with a word of command, naur an edraith ammen!, he thrust the end of his staff into the midst of it. At once a great spout of green and blue flame sprang out, and the wood flared and sputtered.
- The Fellowship of the Ring
Gandalf stood up and strode forward, holding his staff aloft. 'Listen, Hound of Sauron!' he cried. 'Gandalf is here. Fly, if you value your foul skin! I will shrivel you from tail to snout, if you come within this ring.'...
In the wavering firelight Gandalf seemed suddenly to grow: he rose up, a great menacing shape like the monument of some ancient king of stone set upon a hill. Stooping like a cloud, he lifted a burning branch and strode to meet the wolves. They gave back before him. High in the air he tossed the blazing brand. It flared with a sudden white radiance like lightning; and his voice rolled like thunder.
'Naur an edraith ammen! Naur dan i ngaurhoth!' he cried.
There was a roar and a crackle, and the tree above him burst into a leaf and bloom of blinding flame. The fire leapt from tree-top to tree-top. The whole hill was crowned with dazzling light. The swords and knives of the defenders shone and flickered. The last arrow of Legolas kindled in the air as it flew, and plunged burning into the heart of a great wolf-chieftain. All the others fled.
- ibid (Note that he doesn't use his staff to light this fire, he merely speaks a spell)
- He tries (and fails) to open the doors of Moria with the help of his staff.
He stepped up to the rock again, and lightly touched with his staff the silver star in the middle beneath the sign of the anvil. Annon edhellen, edro hi ammen! Fennas nogothrim, lasto beth lammen! he said in a commanding voice. The silver lines faded, but the blank grey stone did not stir...
Again Gandalf approached the wall, and lifting up his arms he spoke in tones of command and rising wrath. Edro, edro! he cried, and struck the rock with his staff. Open, open! he shouted, and followed it with the same command in every language that had ever been spoken in the West of Middle-earth. Then he threw his staff on the ground, and sat down in silence.
- ibid
- Lighting the mines of Moria
Boromir muttered... 'Who will lead us now in this deadly dark?'
'I will,' said Gandalf... 'Follow my staff!"
As the wizard passed on ahead up the great steps, he held his staff aloft, and from its tip there came a faint radiance.
- ibid
- Defending himself in Fangorn
He lifted up his staff, and Gimli's axe leaped from his grasp and fell ringing on the ground. The sword of Aragorn, stiff in his motionless hand, blazed with a sudden fire. Legolas gave a great shout and shot an arrow high into the air: it vanished in a flash of flame.
- The Two Towers
- Scaring everyone in Edoras
[Suddenly Gandalf] changed. Casting his tattered cloak aside, he stood up and leaned no longer on his staff; and he spoke in a clear cold voice. 'The wise speak only of what they know, Gríma son of Gálmód. A witless worm have you become. Therefore be silent, and keep your forked tongue behind your teeth. I have not passed through fire and death to bandy crooked words with a serving-man till the lightning falls.' He raised his staff. There was a roll of thunder. The sunlight was blotted out from the eastern windows; the whole hall became suddenly dark as night... Only Gandalf could be seen, standing white and tall before the blackened hearth.
In the gloom they heard the hiss of Wormtongue's voice: 'Did I not counsel you, lord, to forbid his staff? That fool, Háma, has betrayed us!' There was a flash as if lightning had cloven the roof. Then all was silent. Wormtongue sprawled on his face.
'Now Théoden son of Thengel, will you hearken to me?' said Gandalf. 'Do you ask for help?' He lifted his staff and pointed to a high window. There the darkness seemed to clear, and through the opening could be seen, high and far, a patch of shining sky. 'Not all is dark. Take courage, Lord of the Mark; for better help you will not find. No counsel have I to give to those that despair. Yet counsel I could give, and words I could speak to you. Will you hear them?... I bid you come out before your doors and look abroad. Too long have you sat in shadows and trusted to twisted tales and crooked promptings.'
- The Two Towers
Non-magical Functions of Staves:
1. A walking stick.
Of this Order the number is unknown; but of those that came to the North of Middle-earth, where there was most hope (because of the remnant of the Dunedain and of the Eldar that abode there), the chiefs was five. The first to come was one of noble mien and bearing, with raven hair, and a fair voice, and he was clad in white; great skill he had in works of hand, and he was regarded by well-nigh all, even by the Eldar, as the head of the Order. Others there were also: two clad in sea-blue, and one in earthen brown; and the last came one who seemed the least, less tall than the others, and in looks more aged, grey-haired and grey-clad, and leaning on a staff.
- Unfinished Tales: "The Istari" (describing Saruman, then Radagast, then the blue wizards, then Gandalf)
And:
On the sixth day since their parting from the King they journeyed through a wood climbing down from the hills at the feet of the Misty Mountains that now marched on their right hand. As they came out again into the open country at sundown they overtook an old man leaning on a staff, and he was clothed in rags of grey or dirty white, and at his heels went another beggar, slouching and whining.
'Well Saruman!' said Gandalf. 'Where are you going?'
- The Return of the King
This is crucial - Gandalf has already broken Saruman's "wizard's staff"; Saruman clearly found a long stick and carried it as he walked to the Shire - not for magical purposes, but to help him walk.
'A Balrog,' muttered Gandalf. 'Now I understand.' He faltered and leaned heavily on his staff. 'What an evil fortune! And I am already weary.'
- The Fellowship of the Ring
Again, a wizard uses his staff to support himself.
Gandalf stood in the middle of the span, leaning on the staff in his left hand, but in his other hand Glamdring gleamed...
- The Fellowship of the Ring
And:
Gandalf stood, leaning on his staff, gazing into the darkness, east and west.
- The Two Towers
2. A weapon:
We occasionally see Gandalf whack someone or something with his staff; it would be absurd to do this with a wand.
Gandalf stood before the door of Orthanc and beat on it with his staff.... 'Saruman come forth!'
- The Two Towers
And Theoden's guard, who doesn't know that Gandalf is a wizard, categorizes staves as weapons:
'Follow me!' [the guard] said. 'Théoden gives you leave to enter; but any weapon that you bear; be it only a staff, you must leave on the threshold. The doorwardens will keep them.'
- The Two Towers
Things Gandalf Didn't Use His Staff For:
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. A blinding sheet of white flame sprang up. The bridge cracked. Right at the Balrog's feet it broke, and the stone upon which it stood crashed into the gulf, while the rest remained, poised, quivering like a tongue of rock thrust out into emptiness.
With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished. But even as it fell it swung its whip, and the thongs lashed and curled about the wizard's knees, dragging him to the brink. He staggered and fell, grasped vainly at the stone, and slid into the abyss. 'Fly, you fools!' he cried, and was gone.
- The Fellowship of the Ring (Note that when Gandalf actually engages the Balrog, his staff is gone - he killed a balrog without it)
He raised his hand, and spoke slowly in a clear cold voice. 'Saruman, your staff is broken.' There was a crack, and the staff split asunder in Saruman's hand, and the head of it fell down at Gandalf's feet.
- The Two Towers
- Starting a fire and doing fire stuff to the Fellowship's weapons in Hollin
In the wavering firelight Gandalf seemed suddenly to grow: he rose up, a great menacing shape like the monument of some ancient king of stone set upon a hill. Stooping like a cloud, he lifted a burning branch and strode to meet the wolves. They gave back before him. High in the air he tossed the blazing brand. It flared with a sudden white radiance like lightning; and his voice rolled like thunder.
'Naur an edraith ammen! Naur dan i ngaurhoth!' he cried.
There was a roar and a crackle, and the tree above him burst into a leaf and bloom of blinding flame. The fire leapt from tree-top to tree-top. The whole hill was crowned with dazzling light. The swords and knives of the defenders shone and flickered. The last arrow of Legolas kindled in the air as it flew, and plunged burning into the heart of a great wolf-chieftain. All the others fled.
- The Fellowship of the Ring
Conclusion:
The staff isn't necessary to perform magic, but it seems to help somehow. A wand could probably do this just as easily.
The staff is a symbol of power and authority. A wand could also serve this purpose, but the staff is visually much more impressive.
The staff is handy as a weapon - i.e., for whacking someone with - and a wand isn't.
The staff is also useful for supporting one's weight, and believe it or not, this is what Gandalf uses his staff for more than anything else; a wand would be worse than useless in this regard.
The most likely answer: Staves are just more versatile than wands. They meet a wider array of the wizards' needs.
Note: Part of Tolkien's inspiration for his first wizard - Gandalf - came from artistic representations of "Odin the Wanderer", like these:
Georg von Rosen - Oden som vandringsman (Odin, the Wanderer)
Odin, the Wisdom-Seeking Wanderer by Arthur Rackham
Look familiar? They should. The ideas are very similar: A mighty supernatural being (Olórin the Maia; Odin the god) takes on the form of an old man (Gandalf; "the wanderer") with a walking stick.
