All that a single action trigger does is release the sear, causing the hammer to fall. The weapon must be cocked, either by hand or by the action of the prior shot, before the trigger is operable.
A double action trigger does the work of both cocking and firing the weapon. A double action only (DAO) weapon cannot be fired single action, whereas other double actions can be cocked manually and operate like a single action if so chosen. Many double action semi-automatics (such as the Makarov and others) are double action for the first shot, and single action for subsequent shots.
There are others, such as the Glock "safe action" that are not quite a true double action, but utilize a striker that is partially set by operation of the slide.
As far as "pluses and minuses", that is a whole other story. I own pistols of all types, and enjoy them all.