In both of these cases, the first word is sort of a good thing and the second word is an exaggeration of the first.

To be frugal is to not waste money.  It is a good thing.  But Ebeneezer Scrooge (in Charles Dickens's A Christmas Carol) is beyond frugal.  He is so stingy that he hurts other people.

To be continent is to not overdo things.  It is to do things, but do them in moderation.  To be abstemious is to just not do things.  So if you drink a little bit now and then, you are continent.  If you never drink at all, you are abstemious.