In its original form, rattan is a relative of the tropical palm tree. Rattan is a specific material, but wicker is the general process of weaving rattan or other materials into finished goods.
A rattan chair is definitely made from rattan, but a wicker chair may use other materials such as straw or bamboo slats around a rattan frame. In the 1960's and 70's Dutch Designers Martin Visser and Dirk Van Sleidrecht used the wicker/rattan and rush seating in many of their furniture designs. The company Rohe Noordwolde manufactures a lot of the old rattan.
Martin Visser Low Rattan Lounge Chair with Tubular Chrome Frame
Charlotte Perriand Style Bar Stools
Marcel-Louis Baugniet Bent Plywood and Woven Grass Arm Chair