The time and place are different; the play takes place in Greece at an unspecified time, though Theseus (played in this film by David Strathairn) and Hippolyta (played in this film by Sophie Marceau) are characters out of Greek mythology. The film takes place in Italy during the early 20th century, and we see several items you would never see in ancient times - a Victor Talking Machine phonograph, and several bicycles. But the plot, the names of the characters, and the language of the film remain exactly the same as in Shakespeare's original play. Bottom (Kevin Kline) is given a wife in this version, but she has no speaking lines.
Although Hermia and Helena argue in the original play, there is no mud wrestling.
In the original play, sex between Titania and Nick Bottom is indirectly implied, and we never even see them kiss. This is also true of the 1935 film version. In this 1999 film version, although we do not hear any moaning or heavy breathing, sex between the two is strongly implied. It is also implied that Bottom has an erection when he is with Titania; he looks down at his lower region and is mildly embarrassed, while Titania and the other fairies giggle.
Titania and Bottom kiss several times while lying down.
Near the end, Bottom sees a statue of Titania, recognizes who it is, and gazes fascinated at the statue. Later, just as he is about to go to bed, Titania (in non-human "Tinker Bell" form), hovers outside his window and presumably blesses him. He smiles with tears in his eyes. This is not in the play. In the play, it is never revealed whether or not Bottom remembers Titania at the end, and the two presumably never see each other again.