Asymptomatic means simply "not exhibiting any symptoms at the current time". It does not even even imply (absent context suggesting otherwise) that one is infected, and someone who is infected might be "asymptomatic" at one point and later become "symptomatic" (and hence no longer "asymptomatic"). Or someone (a "typhoid Mary") could have an infection for days, weeks, months, years and remain "asymptomatic" the entire time.
Paucisymptomatic, on the other hand, means having few symptoms ("pauci" meaning "a few"). It does not imply having no symptoms at all. And, as with "asymptomatic", it doesn't imply anything about past or future conditions, only the present state.