First of all: The Q episodes on the Enterprise and the eloquent exchanges with Picard were better than the everyday problems that are on the agenda during Q's visits aboard Voyager. These basically just revolve around relationship and parenting problems.
However, one would imagine an omnipotent pubescent teenager just like Q's son. Basically a kid with a magnifying glass standing next to an ant hill not knowing what to do with his power and his time. A player on the chessboard of the cosmos for whom every sentient life form is merely a pawn that can be sacrificed without guilt to relieve the boredom and infinity of one's own immortality.
The change from Saul to Paul happens a little too quickly for my taste. A spoiled brat whose toys and privileges are taken away doesn't become a saint in a week. Unfortunately, Star Trek also usually falls short in emphasizing their powerful messages. If Q's son had ended up being banished from the continuum as a mortal without any omnipotence and had he joined the Voyager crew and then changed from a bad god to a better human, that would have been a bigger bang. Instead, there is sunshine again in house Q and father and son will travel the universes together in a spirit of adventure.
I don't know why the writers were always so afraid of setting a real, incisive counterpoint. If only episodes like this had been written after Game of Thrones came out. Red wedding and stuff. Have a main character walking the plank. Or confront them with the Q-lessness of life: illness, despair, loss and death. Breaking established rules of storytelling. A mortal Q as a member of the Voyager crew could have been an interesting change (ensign Q*bert for example). Although this then should have happened in an earlier season. Unfortunately, in the end this episode is just another short detour for Q on the Voyager without any long-term consequences.