What is the contrast in generations in Romeo and Juliet (the difference between how Romeo and Juliet and their parents see things)?

The feud is part of Romeo's and Juliet's parents' generation, not theirs.

Romeo and Juliet’s parents continue the feud, but their children are obviously not interested.  Romeo willingly dates a Capulet girl, goes to a Capulet party, and then falls head over heels in love with Lord Capulet’s daughter.  When Juliet finds out that Romeo is a Montague, it doesn’t affect her opinion of him; she is just worried about what her family will think.

We know from the opening scene that Lord Capulet and Lord Montague are heavily invested in the feud, but their wives are not necessarily as passionate about it.

CAPULET

What noise is this? Give me my long sword, ho!

LADY CAPULET

A crutch, a crutch! why call you for a sword?

CAPULET

My sword, I say! Old Montague is come,
And flourishes his blade in spite of me. (Act 1, Scene 1) 

Both Lady Capulet and Lady Montague seem to be complaining about the fighting, but Lord Capulet and Lord Montague get into the fight without much trouble.  Prince Escalus is annoyed by this.  He gives both of them a lecture, separately, which indicates that he thinks that the feud is trickling down from the top.

Romeo has no interest in the feud.  When he sees the aftermath of it, he is saddened.  He says that he wishes people would not fight.  Clearly he is not going to be carrying it on.  He is interested in Rosaline, who is a Capulet.  His friends want to go to a Capulet party, and he goes.  Romeo is not going to be doing any feuding.

When Juliet learns that the boy she met at the party is a Montague, she has a strong reaction.

My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me,
That I must love a loathed enemy. (Act 1, Scene 5)

This doesn’t stop her though.  Later she comments that there is not much to a name.  Romeo’s name is not a part of him.  She doesn’t seem to want to stop seeing him because of who he is.  She doesn’t hate him because of his name.  The feud is something of her family and her parents’ generation, and it means nothing to her.

The fight between Tybalt and Mercutio may be the exception to this.  However, by all accounts, Tybalt has a temper.  He is angry at Romeo for showing up at a Capulet ball. Like the servants in the beginning scene, he wants to make a name for himself by fighting and uses the feud as an excuse.  This makes Romeo a prime target.  Tybalt dies because he's a hothead.

After their children die, Lord Capulet and Montague bury their feud.  They realize that it is not worth losing their loved ones over.  It takes their children killing themselves for them to realize how wrong they were.