There's a world of difference between cocktails and mixed drinks. It’s not only in the number of ingredients, but in preparation, intention, and even philosophy.
Mixed drinks are those with two ingredients, one of which is alcohol, such as gin and tonic. Cocktails are more complicated; typically, there are at least three ingredients.
The golden ratio in cocktails mandates a base liquor combined with sweet and sour flavors. The complexity is reflected in more sophisticated preparations like shaking and stirring.
A cocktail includes more ingredients and is more time-consuming and challenging to prepare. It's a balancing act of complexity.
A mixed drink, by contrast, simply has liquor and something else: generally, soda or juice. It's uncomplicated and has no specific recipe.
Cocktails are made from recipes to achieve harmonious flavors. The mixer in mixed drinks may be used to mask the strong taste of the base liquor.