The difference is slight. A chef is someone with great cooking ability and skill.They can cook gourmet (fancy/expensive/difficult) food. Rather, a cook is more a generalist. You and I can both cook, and be both good and bad; but, unless we are particularly skilled we are not really chefs.

A chef is someone that has trained as a cook, usually they are trained at a special culinary school. A cook is someone that is not trained.

There is a joke in the English-speaking world, a chauvenistic joke, that says that only men are chefs...women are cooks. It implies that higher level gourmet etc is done by men - the best food, and that cooking, general food preparation, is done by women. Of course this is not true, my ex-girlfriend was terrible at both :-) Kemi and Sonia above "hit the nail on the head" - (idiom - they got the explanation exact).

This is what you're looking for: http://www.differencebetween.net/miscellaneous/difference-between-cook-and-chef/