Luhrmann and Zeffirelli have different interpretations of Romeo and Juliet because they have different goals. Zeffirelli is trying to produce a classic Shakespearean production, much like you would see on stage. Luhrmann is trying to produce a marketable, popular version of Romeo and Juliet.
The most obvious differences between the two movies are the settings and props. Neither movies stay completely faithful to the text in this respect. The Luhrmanm version strays from the text by changing the location from Verona, Italy to Los Angeles, USA. The props are then updated from the 1600s to their modern-day equivalent: for example, the guns are called "swords" and the coastal area is named "Verona Beach." Even though Zeffirelli chooses a more classical setting, he still takes some liberties with the scenery. Mostly these are minor changes in outdoor scenes that couldn't be literally portrayed on a stage. Some of these scenes, like Romeo and Balthasar galloping from Mantua, weren't even in the written text, and others, such as the large peasant-filled Verona square, could be depicted less extravagantly.
Another significant difference in the directors' interpretations of Romeo and Juliet is what they leave out or edit. The murder of Paris is totally cut out of both movies. This single abridgment totally alters the ending of the story. In Shakespeare's version, Romeo still could have been accepted into society before killing Paris --as Father Laurence pointed out--, since he was exlied for killing Tybalt, Mercutio's murderer. In Both movies Romeo chases Tybalt and challenges him to a duel, but in the original it's Tybalt who attacks Romeo. Romeo was defending himself when he killed Tybalt, so he's in a good position in the eyes of the 16th century law system. In the movies, Romeo's suicide is even more of an act of blind passion than it was in the original text.
As well as leaving out Paris' death, the Luhrmann production destroys the ending scene. By having Juliet wake up while Romeo is dying the mood plummets from depressing to the pit of despair. As if things weren't bad enough, the Capulet and Montague lords don't reconcile with each other in the light of their children's death. The last little bit of hope is beaten, maimed, and thrown to the dogs.
Both the Zeffirelli and the Luhrmann show how bizarre Romeo and Juliet really is. It's stuffed full of sex, violence, murder plots, family feuds, conspiracies, and any number of other illegal activities. These things are much easier to spot in the Luhrmann version because of its modern setting. We would never expect two people who start a gun fight at a gas station to be let off with a mere warning from the police. Because of the remote place and time it's occasionally possible to lose sight of how strange the story really is when reading the text or watching the Zeffirelli production.