"Do it very quickly" vs "do it ASAP"

What is the difference between these phrases?

Please, do it very quickly, since the deadline is approaching.

and

Please, do it ASAP, because the deadline is approaching.

Even though ASAP stands for "as soon as possible", I think it has come to take on the special meaning of "this is your highest priority; drop everything and get it done NOW". At least, that is what I believe it means in a typical American workplace.

Any other phrasing, including "very quickly" or even "as soon as possible" spelled out, is implicitly less urgent.

To do something soon means to minimize the length of time elapsed till the action begins. To do something quickly means to maximize the speed at which the action takes place. Of course, this is a somewhat literal interpretation.

My interpretation is ASAP means complete the task with high priority and very soon, but it does suggest some leeway to account for other work that the person saying it isn't aware of. 'Quickly' feels more like you're implying the task is already underway and that the length of time spent on the task might be of greater importance than getting it done right. Both of these are highly subjective opinions though.

A certain type of person (boss, superior...) might also use ASAP in a more passive aggressive way, without regard for other considerations and imply that there task has the absolute highest priority. It could be used as a form of fake nicety in a minor of cases.