Wicker is a process (not a specific material) of weaving any number of materials such as reeds, bamboo, rush, straw and willow into a finished piece of furniture. The confusion comes about because this weaving generally takes place over a Rattan Core frame.
Rattan itself as an actual vine, which grows in tropical forest regions of Africa, Asia and Australasia. Rattan grows in a solid pole shape hundreds of feet in length making it durable and harder to break than bamboo, which is hollow. The outer skin of the vines (the Peel) are stripped and used as a rattan weaving material to wrap furniture joints. The Core is then left to use within the frame.
Some manufacturers will even use the phrase "cane furniture" in place of either rattan or wicker to try and avoid this confusion.
To try and make this simple, if you buy Rattan Furniture it will probably have a Wicker Weave on top and if you buy Wicker furniture it will probably have a Rattan Core.
The best tip I can give is, "always ask before you buy".