We tend to use don't in speech and informal writing. In formal writing, it is better to use do not. This form would also be used in legal documents and official signs - DO NOT ENTER. Do not open this door. You would mainly use do not in speech if you wanted to put emphasis on the imperative - Do not go out without turning the iron off. A parent telling a child off, or someone giving instructions might use do not in speech. Otherwise, don't is more common.

Don't is just an abbreviation of the words do not. They mean the same thing and are used in the same context. eg: "Do not go that way" is the same as "Don't go that way". English native speakers tend to use the word "don't" as it is quicker to say. And yes you can always write "Don't".