The biggest difference between Puck and Bottom can be seen in their levels of worldly intelligence. Puck very rightly turns Bottom's head into the head of the proverbial ass because Bottom is indeed acting like a donkey in terms of being an idiot. Due to Bottom's enthusiasm for the play and for his visions of himself as a performer, Bottom very idiotically asks to play all three lead roles of the play, Pyramus, Thisbe, and the lion. This is particularly idiotic because it will certainly be impossible and ridiculous for Bottom to play both lovers acting on opposing sides of the wall. It will equally be ridiculous to play a lion that scares himself when he is also playing Thisbe.
Puck, on the other hand, is actually very insightful and intelligent. Puck very rightly observes the ludicrousness of the four lovers' response to having their minds meddled with in the woods. Puck observes that it is absurd for Helena to instinctively disbelieve the sincerity of both men, and it is equally ridiculous for her to accuse her best friend of being in on the joke. We see Puck's wise response to the lovers' fight in the woods in his lines, "Shall we their fond pageant see? / Lord, what fools these mortals be!" (III.ii.115-116).
The one similarity between Bottom and Puck is that they actually both resemble the proverbial ass in that they are both equally stubborn. We see Bottom being inadvertently stubborn when he persists in his foolish ideas concerning the play. We see Puck being stubborn in his persistence of performing mischievous antics. While Puck claims that mixing up the Athenian men was a mistake, we know that Puck is notorious for his pranks. We see Puck claim his innocence in his lines:
Believe me, king of shadows, I mistook.
Did not you tell me I should know the man
By the Athenian garments he had on? (III.ii.365)
We also learn that Puck is a notorious prankster when we first meet him because his fellow fairy describes him as a "knavish sprite," meaning "mischievous spirit" (II.i.33).