Cook: A person that has no proper training or attend to any culinary school. They have no or very limited skill, experience or knowledge about kitchen. They can be anyone including out from the street or anywhere you can think of.
Chef: A proper trained & educated cook in a proper culinary school. Have basic skill, experience & knowledge about how the kitchen works & knows how to manage the kitchen. They must get out of a proper culinary school to earn the title of chef.
Bottom line is:
A chef is a cook but a cook is not a chef