The main difference between “cannot” and “can not” is that the first term deals with impossibility and the second handles choice. The two-word version also is more common when a person wants to add emphasis on something, or when he wants to stress that someone has more than one skill. Mistakes about how to use them properly means that they are virtually interchangeable in regular speech, however.
Impossibility vs. Choice
One of the most common explanations for the difference between “cannot” and “can not” is that, with the former word, there is no chance someone can carry out an activity, no matter how much he might want to do so. The following sentences both are based on this concept of impossibility:
Jane cannot jump 50 feet in the air.
We cannot go to Pluto.
In these examples, “cannot” essentially means “lacks the ability to." In the first instance, for example, the way the human body is made and functions always will keep Jane from reaching the extreme height, regardless of her fitness level. The second sentence is true because currently, scientists do not have all the knowledge or technology that a mission to Pluto would require.
By contrast, when a person uses “can not,” the ability to do the action still exists.
Yuri can not study anymore.
Sam can not have a drink.
For both these cases, “can not” translates to “is not going to” or “is opting not to.” In other words, the subject has a choice. Yuri can study, for example, or he can go do something else. Sam can consume a beverage, or he can leave it be.
Can’t
Typically, a contraction represents the joining of more than one word, usually as a way to make speech shorter and more efficient. Contractions also can eliminate syllables within words, however, sometimes for a poetic purpose, such as sticking to a particular meter. A good example is “e’er,” a shortened form of “ever” pronounced like the words “air” or “heir.” Most experts say that the contraction “can’t” functions in this way and, therefore, is a shortened version of “cannot” rather than a joining of “can” and “not.”
With the contraction representing “cannot,” technically, someone only should use it when they really mean that the subject of their sentence does not have the ability to do something. In common speech, however, people frequently use it incorrectly. A person might say, "I can't cook tonight," for example.
Emphasis
The word “cannot” rarely needs to receive emphasis because the meaning is not ambiguous — it always means that it isn’t possible for the subject to complete the action. The phrase “can not,” however, deals with options, and people can stress what they want to happen or are going to pick. A father upset about his daughter’s short skirt, for instance, might say, “You can not wear that out of this house, young lady!” He essentially would be stressing that, even though she’s physically capable of donning the offensive clothing and going somewhere in it, he’s not going to let her, making her choice for her. In the same way, when a person says “You can not be serious,” he really means, “You are choosing not to be serious,” which is a roundabout way of expressing disbelief or disapproval and saying he wants the other person to quit fooling around.
Multiple Abilities
Related to the idea of emphasis is the concept of stressing that someone has multiple abilities. People usually do this using “not only…but” statements, such as “He can not only sing, but he acts, too.” A person generally can rewrite a sentence like this to make the dual abilities a little more clear, such as “He can sing, and he can act, too.” The first construction, however, draws a little more attention to the second ability, essentially saying, “Yes, he has Ability A, but don’t ignore Ability B.”
Commonality
Even though these two terms technically do have slightly different meanings, widespread inappropriate application has made them almost interchangeable in everyday language. According to some sources, though, “cannot” is the more common version. Location has some influence on usage, with some areas preferring the single word and others preferring the phrase.
