Osmosis and diffusion, while related, are not exactly the same thing. Diffusion is defined as the movement of molecules from an area of high concentration to an area of low concentration, and osmosis is defined specifically as the diffusion of water across a semi-permeable membrane.
While both processes involve the movement of molecules down a concentration gradient, the difference lies in the fact that the term "osmosis" refers specifically to the movement of water molecules, while diffusion can involve molecules of any type. Both osmosis and diffusion are spontaneous processes, which mean that they occur without the input of any outside energy. The term osmosis only refers to the movement of water in the liquid state. However, the term diffusion is used to describe movement of molecules in either the liquid or gaseous state. For example, if carbon dioxide gas is released in the center of a room, it slowly diffuses throughout the room until the concentration of carbon dioxide is uniform across the room.
The word osmosis is most commonly used in the fields of cell biology and physiology. When a cell is placed in an hypertonic solution, or one that contains a higher concentration of solute than is found inside the cell, water moves out of the cell spontaneously, and this is called osmosis. If a cell is placed in a hypotonic solution, or one that contains a lower concentration of solute than is found inside the cell, water flows spontaneously into the cell, and this process is also referred to as osmosis. In both of the cases described above, one could also say that diffusion of water occurred across the cell membrane.