At least in terms of cooking, what varies is the speeds and types of ingredients. Mixing is used when referring to dry ingredient, like mixing spices or mixing a combination of dry and liquid ingredients. You usually only mix until the ingredients are all evenly spread Stirring is used when you are mixing two liquid ingredients or a liquid and a dry ingredient. Stirring usually occurs for a longer period of time than mixing as the goal of stirring is to keep the mixture in motion until a cooking goal is accomplished (the ingredients are cooked thoroughly without sticking or a specific texture is achieved). Whisking (or whipping) is a very fast stirring, usually with a tool that has several thin tines. The goal of whisking is to aerate the mixture thereby increasing it's volume. It's how merengue or whipped cream is made.

To stir means to move something (a spoon or electric mixture) in a repeated motion to combine a number of things together. Mix means simply to combine different elements together. It might or might not involve using a motion of a tool to do it. I could mix languages together. It could be real physical mixing or intellectual mixing. To whisk means to stir with a fast motion using a wire mesh. Some cake mixes or other recipes require this. [Note: whisk also means to wipe something away with a quick motion of the arm.] To review: If I whisk to combine things, I am stirring and mixing. If I stir, I am mixing, but not necessarily whisking. If I mix, I may be stirring and/or whisking, but not necessarily.