The difference between wicker and rattan is the distinction between a product versus a process. Rattan is a type of wood with a vertical grain that is often used to make woven furniture items. An item made with a particular weaving style — whatever material is used — is called wicker.
In its original form, rattan is a relative of the tropical palm tree. It starts to grow upwards like a tree, but then bends back to the ground and snakes through the rain forest like a vine. After a few years of growth, the vines are cut into 12 to 18 foot (about 3.7 to 5.5 meters) sections and hauled away for drying. Furniture companies that specialize in furniture made with this plant often have processing plants in the Philippines or Southeast Asia to treat the wood and ship it out to large furniture manufacturing plants in America. North Carolina has a significant number of wicker and rattan furniture companies.
Rattan is considered to be one of the strongest woods available, since its grain grows vertically instead of forming the concentric rings of most other hardwoods. The straight wood is usually steamed and then bent into the desired shape through the use of specialized shapers. Once dried, it will retain its shape forever. These poles are often used to form the frames of what will become rattan or wicker furniture.
Here's where the difference between the two terms lies: rattan is a specific material, but wicker is the general process of weaving this wood or other materials into finished goods. A rattan chair is definitely made from that specific wood, but a wicker chair may use other materials such as straw or bamboo slats around a rattan frame.
There is no specific material called wicker, and some manufacturers may even use the phrase cane furniture instead. It is essentially the same thing as wicker, but the material used in the weaving may or may not be rattan.
Rattan vines may also be peeled mechanically to form thin slats for weaving. The curved outer layer is also used as a decorative trim to hide the rough seams created by the wicker process. Bamboo may also be used for trim, but rarely as a form. Bamboo is hollow, which means it often cracks when steamed and bent. Rattan is solid, which makes it an ideal material for lightweight but solid furniture.