Compare Romeo's love for Rosaline with Romeo's love for Juliet.
Romeo is smitten with both Rosaline and Juliet, but his love for Rosaline is unrequited. Thus, we find him at the beginning of the play pining for Rosaline, who has decided to remain chaste. His friends urge him to get over Rosaline, but he only does so when he encounters Juliet. The Friar, to whom he has spoken at length about his love for Rosaline, is skeptical about his newly-professed love for Juliet. He understandably thinks that Romeo has impulsively given his love to another, which suggests that neither love was really sincere. It is hard to know whether his love for Rosaline was legitimate—it certainly seems so at the beginning of the play—but his love for Juliet certainly is. He is willing to die for her. I would argue Romeo loves both women passionately, but his love for Juliet seems more substantial.
When we find Romeo at the beginning of the play's action, he is the very picture of a love-sick youth, who (according to his parents) routinely shuts himself out from daylight and spends his time crying and bemoaning his misfortune. This state of melancholy, we quickly discover, has been brought on by his apparent all-consuming (unrequited) love for a young woman named Rosaline. It is noted fairly early on that Rosaline has taken a vow of chastity:
"Well, in that hit you miss: she'll not be hit
With Cupids arrow, she hath Dian's wit"
In this case, "Dian" alludes to Diana, the Roman Goddess who, along with Minerva and Vesta, were maiden Goddess' who swore to never marry. From this revelation alone and Romeo's reaction to it, we see several elements of human behavior at play, behavior typically exhibited by a young male teen. For starters, more than anything else, Romeo references Rosaline's physical beauty when describing her, that it is without compare and that he is blind to the beauty of other women because he has already witnessed the fairest woman in existence. So, whenever it came to light that Rosaline--the object of young Romeo's affection--planned to take a vow to never marry, the very thing that Romeo claimed to have wanted most was the very thing he could never have. Combined with a predictable amount of hormonal inspired lust and you have a perfect cocktail for world ending, self-indulgent teenage despair.
At this point, Romeo displays a lack of ability to take action to change his current state beyond moaning and cursing, behavior which is directly contradicted later, when he meets his true love, Juliet. However, this early in the play, Romeo's understanding of love is very shallow, no doubt inspired by the poems and romantic literature he shuts himself up with, as evidenced by the impromptu poetry occasionally recited to his cousin and best friend Benvolio. In summary, Romeo feels that his intense affection and lust for Rosaline will never be satisfied, that he "lives dead" to tell his story. Although I believe Romeo's feelings of sadness and melancholy to be sincere, I also detect several hints in the text of self-awareness within his supposed struggle and that some part of him finds his suffering to be a truly romantic thing. In other words, Romeo is more in love with the idea of love but has not found it yet.
Act 1.5, enter Juliet. This is where everything actually changes for Romeo. Upon seeing her he exclaims that "she doth teach the torches to burn bright!" and in one of the most famous lines from the play, asks himself:
"Did my heart love till now? Forswear it, sight,
For I ne'er saw true beauty till this night."
See what he did there? Now, this declaration of love may seem no more profound than before due to the fact that he more or less said the same thing (albeit less poetically) about Rosaline earlier on. However, where Romeo really demonstrates to the audience that something new has been awoken inside of him is what he does next. Trying not to blow his cover as a Montague, he sneaks up to Juliet during the party and touches her hand, which he refers to a holy shrine made dirty by his rude hand. Their subsequent interaction pours out of them in 14 lines of perfect iambic pentameter ending in a rhyming couplet, making it a sonnet. For God sake, it's their first time speaking and they unintentionally wrote a SONNET together! This interaction ends with the first kiss of the "star-crossed lovers" and seals their fate. Romeo knows in this moment that he did not truly live until now and that their is no going back. Unlike with Rosaline, he makes good on these claims time and time again throughout the story, putting his life in very real danger in order to be with her, including the famous "Balcony Scene" of act 2.2, where Romeo climbs a high wall to sneak into the Capulet garden. If caught, he will almost certainly be tortured and killed but Romeo is determined, telling Juliet that his life were "better ended by their hate, Than death prorogued, wanting of thy love." A new Romeo has been born and the true action of the play is initiated at this point.
All in all, it is Romeo's actions that should illuminate the difference between his love for Rosaline and Juliet. To borrow a line from another famous character of The Bard:
"What wilt thou do for her?"