"Will" has the connotation of a choice to perform the action. "Do" more simply centers around whether the action happens.
Taking the prefatory clause "If you will not leave tonight", the connotation is that the subject of this clause can decide whether he "does" or "does not" leave. The clause "If you do not leave tonight" is similar overall but has the connotation that whether the subject wants to leave or not may not be the final say in the matter.
Overall, the difference to American English speakers is typically that "will" is used to emphasize that it's the person's own choice and nobody else's whether the action is performed, and thus the hidden meaning is usually "What happens next is all on you", while "do" is more neutral, implying no choice at hand, so the meaning behind it can more easily be "the future depends on this condition but I know it may not be completely up to you".
For this reason, "do" is heard much more often in American English because it's more polite-sounding, implying that the present or future situation is not necessarily the result of the subject's choice in the matter.