What are the similarities and differences between Puck and the god Pan in A Midsummer Night's Dream?

This well-loved play by Shakespeare contains a number of characters who are fairies that live in the forest. Puck seems to be their leader and is a mischievous, bawdy character who has inspired countless works of art and literature. Comparisons of Puck to Robin Hood (his other name in the play is Robin Goodfellow) have been made, but Puck is also similar to the Greek god Pan, ruler of the forests. Like Puck, Pan has a reputation for being mischievous. Pan is also associated with the more carnal aspects of sexuality (in part due to his being half goat, from which the expression "horny" derives), and Puck's dialogue contains many bawdy sexual innuendoes, as in this speech:

"The wisest aunt, telling the saddest tale,
Sometime for three-foot stool mistaketh me;
Then slip I from her bum, down topples she..."

William Blake's painting "Oberon, Titania and Puck Dancing with Fairies" portrays Puck much like the satyr/god Pan, with horns and a crown of leaves. On Pinterest (which can be a very good source for finding images of artwork of all historical periods), page entitled "Puck" reveals many portrayals which characterize Puck as a satyr, like Pan. The satyr, a mythological being with the upper body of a human and lower body of a goat, with horns, is often portrayed in art playing pipes (the are also known as "Pan pipes"). Pan's realm of Arcadia (the woodland paradise of the gods) also gives him leadership over the pastures, giving him a shepherd aspect, reflected also in his goat like appearance.

Puck's name is said to be a version of "Pwca" (pooka) which comes from the Welsh word for a hobgoblin, a mischievous fairy who plays tricks. Sometimes the pooka is portrayed in folklore as quite malevolent. Puck is usually portrayed with good humor whose pranks are meant in fun; in this way the appearance of the fairies in the play are occasions of humor. But Oberon, the King of the Fairies, is malicious and tries to hurt Titania in retaliation for her actions. In Greek mythology Pan is the equivalent of a king of the forest; but in the mythology of this play, Puck is more of an underling character, in servitude to Oberon.