These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Ear Worm: The entire soundtrack, special mention to the infamously catchy "Lovefool".
Ensemble Darkhorse: Pete Postlethwaite as Father Laurence, due in large part to his actually knowing how Shakespearean dialogue works.
Faux Symbolism: Water as the overarching motif (the first shot of Juliet is of her face underwater, the lovers meets across an aquarium, the balcony scene is reset in a swimming pool, Tybalt dies after toppling into a fountain, etc), along with ubiquitous Catholic symbols. And even the director is not quite sure what all of it is actually supposed to mean.
Ham and Cheese: Whether you love it or hate it, the actor playing Mercutio was having way too much fun in his role.
Romeo's reaction to Mercutio's death makes the scene all the more heartbreaking - he holds his best friend's body to his chest, sobbing in impotent fury. Then he stands, holds his head in his hands, then slowly starts running towards his car, ignoring Benvolio's pleads to stop...
For that matter, look at Tybalt's face in that moment. As his anger subsides, he looks soberly at what he's done, and you almost want to give him a hug. His life of passion and mayhem just imploded by his own hand, and you can see in his eyes he's stuck somewhere between sheer panic and genuine remorse. Mercutio wasn't who he hated or wanted to see dead, and while him bolting doesn't win him a whole hell of a lot of sympathy, he clearly regrets the deed.
The ending, which wouldn't have happened if Romeo had just looked down and seen Juliet waking up.
As the camera slowly pans outwards in the chapel of rest scene, with the unbearably sad Liebestod from Wagner's 'Tristan und Isolde' playing as we're shown clips of the pair in happier times.