In Romeo and Juliet, what are the differences and similarities between the characters of Romeo and Juliet?   

I'm sure you'll get plenty of responses to this question, as there is so much information in the play that can lead audiences to compare the two characters.  Foremost, they are both young, from families that are "both alike in dignity," and they are both capable of selfless love. 

In Act 2, scene 2 (the balcony scene), we see evidence of a difference between the two.  Here, Juliet is a little more level-headed than Romeo, who is easily carried away by his poetic and idealistic views on romance.  When Romeo starts to make a promise to Juliet ("Lady, by yonder blessed moon I vow..."), she chides him:

O, swear not by the moon, th' inconstant moon

That monthly changes in her circled orb,

Lest that thy love prove likewise variable.

Essentially, Juliet recognizes that while it might sound like a good idea to swear by the moon, the moon is changing and inconstant.  So, Juliet tells Romeo not to swear by the moon unless his love will change like the moon does.

When Romeo persists, Juliet tells him that she is wary of the relationship:

Although I joy in thee,

I have no joy of this contract tonight.

It is too rash, too unadvised, too sudden,

Too like the lightning, which doth cease to be

Ere one can say it lightens.

Again, these lines show a difference between Romeo and Juliet; she is a little more cautious than he is at this point in the play. 

In terms of how they use language, they are very different too.  A good example of this is the way we first meet both of them.  Romeo is lovesick and spends many words in his opening scene with Benvolio describing love and how "lovesick" he is. He says:

Love is a smoke made with the fumes of sighs;

Being purg'd, a fire sparkling in lovers' eyes;

Being vex'd, a sea nourish'd with lovers tears;

What is it else?  A madness most discreet,

A choking gall, and a preserving sweet.

In contrast, in her opening scene, Juliet has only two lines, one of which is in response to her mother's and the Nurse's invitation to check out the very eligible bachelor, Paris, whom she will meet at that night's party.  She doesn't seem interested in Paris or falling head over heels for him.  She says:

I'll look to like, if looking liking move,

But no more deep will I endart mine eye

Than your consent gives strength to make it fly.

So, while Romeo is consumed by love, its highs and lows, and going on and on about it with Benvolio, Juliet is much less talkative and seems (before she meets Romeo) not at all interested in love.