Getting something done soon simply means within a short time.  This is, of course, a subjective measure and arguably depending on your perspective could be minutes, hours, days, months, etc.

The term is relative and so if the task would normally take a century, the term soon could refer to getting it done in half a century.  That is earlier (or soon) relative to the normally expected time.

Sooner

The comparative adverb of soon is sooner.  While still a subjective measure there is an implied baseline.  In essence, the expectation is the task would be completed before the initially selected duration.

Logically though, with the term sooner there is a bit of a paradox because any moment of time before another moment of time is sooner so the soonest time does not exist because there is always a sooner time (considered on an absolute basis).

Getting it done earlier

Finishing an activity or achieving a goal before the accepted duration could be sooner but it is not necessarily faster.  Doing it sooner is about reduced duration.

It can be achieved by doing less work but still attaining the same outcomes.  That is working smarter not harder.   Getting it done sooner does raise the question as to why it was not scheduled soonest in the first place.

This is not a question I ask when dealing with bureaucracy.  There is really no point.  The culture in these situations is largely focused on process versus urgency.  (In many cases mindless process).