The wand, like the athame, is a tool used to channel and direct the energy from the magic user to a particular purpose. However, the energy from the wand is a gentler and more subtle use of energy. It is used to "invite" rather than to "command". Thus one might use the wand when evoking Deity into one's circle. The wand, at least one made of wood, is also used in dealings with the Fae or Fair Folk as they do not tolerate the presence of iron.
The staff is for a coven what the wand is for the individual Wiccan. While an individual can use a staff, the staff is generally reserved for group workings.
The staff and the wand hold Elemental correspondences. In the Tarot and Ceremonial magic they are associated with the Element of Fire. In other Traditions they may be associated with Air.
Both the wand and the staff are used to gather and direct energy. The magic user gathers up the energy for their working and directs it through the tool to accomplish their objective. It is thought that this energy is less commanding than the energy directed through an athame or sword...more of an inviting. Traditionally the wand or staff is used when inviting any of the Fae to enter a circle.
Traditionally both the wand and the staff are made of wood. (Some make them of metal but I'm not fond of this as it defeats some of the necessary energy of a wand.) Different woods have different magical properties and you need to keep this in mind when selecting the wood for your tool. There is some excellent information on the symbolism of different trees at http://www.dragonoak.com/Magical-Wood-Properties.html My personal wand that I use most often is made of 500-year-old Irish bog oak and elder, uniting both the male and female properties in one wand. I also have one made from the apple tree in my backyard.