Paris goes to Juliet's father when he decides he wants to marry her, which shows he's not very passionate and pretty conventional. Romeo, in contrast, goes right to Juliet herself; he stands beneath her balcony and convinces her to love him (not that she needs any convincing). Then, when Juliet dies, Paris goes to her grave to scatter flowers and cry. In contrast, Romeo goes to her grave to commit suicide because he can't live without her.
Paris seems to hang out a lot with Juliet's father, and he never gets into trouble fighting on the streets. This makes him seem boring in contrast to Romeo, who hangs out with a posse of rowdy friends who are always getting into huge public brawls.
Paris asks politely to marry Juliet but never really explains why. He assumes that Juliet must be in love with him. Romeo can't stop gushing about Juliet's many perfections, but he's nervous and never assumes that Juliet loves him until she tells him so. Paris kisses Juliet only once, in front of the Friar, and then he gives her only a "holy kiss," probably on her cheek. Romeo kisses Juliet within a minute of meeting her. Paris never has sex with Juliet; Romeo does.
It's all in the language. Contrast Paris and Romeo's reactions at Juliet's tomb, as scholar Marjorie Garber does, and it's painfully obvious that Paris is lacking in passion."The obsequies that I for thee will keep / Nightly shall be to strew thy grave and weep," he says (5.3.2). It's like a nursery rhyme, and about as romantic. Romeo, in contrast, is furious over Juliet's death, and eloquent in his fury. "The time and my intents are savage-wild / More fierce and more inexorable far / Than empty tigers or the roaring sea," he says (5.3.1). There's just no competition. Romeo has all the good lines; nothing Paris says is remotely memorable.
Many of Romeo and Juliet's most romantic lines are spoken in "blank verse," which, ironically, is not blank so much as very structured: 10 syllables, 5 beats, with the emphasis falling on every second syllable. (It's called iambic pentameter.) As Juliet says, "deNY thy FAther AND reFUSE thy NAME / or IF thou WILT not BE but SWORN my LOVE / and I'LL no LONGer BE a CAPuLET." Of course, actors reading these lines shouldn't talk in a singsong voice. But the rhythm underlies what they're saying and makes it more poetic. It also means that Romeo and Juliet can literally finish each other's lines, which shows how in tune they are with each other. You can tell when characters are speaking in blank verse because the lines on the page will often be in a long skinny column (one line = 10 syllables). Prose is in normal paragraphs that go all across the page.
Not everyone in the play speaks in blank verse, which is the elegant, high-class way of talking. People lower on the social scale – like the Montague and Capulet servants who open the play – don't talk in a special poetic rhythm. They just talk. Contrast some of the Nurse's speeches or Mercutio's dirty jokes – normal language – with the passages of Romeo and Juliet's romantic blank verse.