"Cannot" means "It is impossible." If you say "I cannot go to the store", you're saying that it is literally not possible for you to go to the store. Maybe you're at work and aren't able to leave, maybe the weather is so bad that it keeps you in your house, maybe you're too busy with other things to get away.
On the other hand, "I can not go to the store" means that it is possible for you not to go the store. You could go to the store if you wanted, but if you don't want to go, you don't have to go. Nothing is requiring it.
"Can't" is always a contraction for "cannot". There is, as far as I know, no contraction for "can not". Which is just as well, because this usage is pretty rare.
This mini-rant brought to you by the sign in my parking garage which contains this error*, and drives me batty every time I notice it.
*"Cars left after closing hours can not be retrieved until the next business day." Which, taken literally, means that you can choose to leave your car until the next business day. I have a feeling that this is not what the sign-makers had in mind.