Puck and Bottom are nearly as different as day and night, but one similarity is that both have a compassionate and sensitive nature.
We see Puck's compassionate and sensitive nature when we see that he is moved by what he perceives to be Demetrius's hatred of fair Helena. Oberon, having seen Demetrius scorn Helena in the forest and seeing how love sick she is, tells Puck to apply the flower to Demetrius's eyes so that he falls in love with Helena, righting a grievous wrong. Oberon tells Puck that he will recognize Demetrius by the "Athenian garments he hath on"; however, neither Oberon nor Puck realize that there are actually two Athenian men in the forest that night; therefore, Puck mistakes Lysander for Demetrius, making Lysander fall in love with Helena instead of Hermia. When Puck finds whom he thinks is the correct Athenian man and sees how far away the lady is sleeping from him, having been rejected by him and also sees how beautiful the lady is, he feels great pity for her. Puck expresses his anger towards the man by calling him a villain, as we see in his lines, "Churl, upon thy eyes I throw / All the power this charm doth owe" (II.ii.78-79). When we see Puck call the man a "churl," or "villain," we know how much compassion Puck is feeling for the lady and see that Puck has a sensitive and compassionate nature.
We likewise see that Bottom has a very sensitive nature. We especially see this when he expresses his concern that having Pyramus slay himself with his sword will offend the ladies in the audience and also his concern that the ladies will be frightened by the lion in the play, as we see in his lines, "Pyramus must draw a sword to kill himself, which the ladies cannot abide" and, "[A] lion among ladies is a most dreadful thing" (III.i.9-10, 28-29).
However, while both Puck and Bottom have caring and sensitive natures, one major difference between them is their level of intelligence. Puck is Oberon's court jester and renowned prankster. Every witticism he makes is intelligent, and he makes witticisms intentionally. One good example of this can be seen in the first line of his closing monologue in which he mocks Quince by repeating Quince's opening lines of the prologue he delivered earlier. For instance, Puck's line is "If we shadows have offended" while Quince's line is, "If we offend, it is with our good will" (V.i.418, 115). In contrast, Bottom comes up with some very idiotic suggestions for the performance of the play. In particular, since he believes that the ladies will be bothered by a gruesome death and by the presence of a lion, he proposes that they explain that the events of the play are not actually real in a prologue, as if an audience of a play would not already be able to figure that out for themselves.
Hence, we see that while Puck and Bottom share a sensitive and caring nature, they greatly differ in intellect.