everyone wants in the play, and while there is no "bet" placed in the actual play, Kate stands in the way of all of Bianca's suitors. The eldest daughter must marry before the youngest or face a life of spinsterhood. Therefore, Bianca's suitors attempt to arrange for Kate's marriage as a means to get to Bianca.

The movie is actually fairly closely tied to the play. Most of the differences are simple modernizations in order to make the movie "work" for the younger generation. Probably the most obvious of these differences is that the story is set around getting a date for the prom instead of getting married. Another difference is in the characterization of Kate. In the movie, Kate paints a picture of a rough relationship in the past that has lead to her recent disillusionment with boys and dating. In the play, we just see Kate as a naturally mean spirited shrew when it comes to dealing with people. One other main difference is in the relationship between Kate and her final mate. In the movie, Kate has definately fallen for the boy and we're lead to believe there will be a happily ever after. In the play, we don't know for sure if Kate is really in love (or even really likes) Petruchio or if she is simply obeying his will in order to avoid further taming.

This question could  make quite an extensive essay, so I'll cover two relationships and point you, hopefully, in a useful direction.  Also, since you don't distinguish, I'll be using the characters from 10 Things, the TV show.

The first major difference between Kate and Petruchio in Shrew and Kat Stratford and Patrick Verona is the way they interact.  In the play, Petruchio determines to make Kate his wife, regardless of how she feels about it.  He gives the reason of money:

...and, therefore, if thou know

One rich enough to be Petruchio's wife...

She moves me not...

I come to wive it wealthily in Padua. (I, ii)

But he also seems excited by the challenge posed by the wildcat Kate.  So, he meets Kate and directly confronts her and tells her she will be his wife, whether she likes it or not:

Thus in plain terms:  your father hath consented

That you shall be my wife; your dowry greed on;

And will you, nill you, I will marry you. (II, i)

So the die is cast early in the play, and Kate is resistant the whole way.  When Petruchio says that they'll marry on Sunday, she retorts:

I'll see thee hang'd on Sunday first. (II, i)

While Kate and Petruchio, from the moment that they meet are struggling over the questions of first betrothal and later appropriate behaviour between husband and wife, Kat and Patrick are simply students in high school, who are attracted to each other, but always seem to be at odds over something.  This sort of cat and mouse tension works well for a TV show that must run over many episodes, and also displays the awkwardness  that can exist between teenagers when it comes to love.  They don't come anywhere near the life-changing topic of marriage, and are pretty much confined to the common issues facing teenagers in today's society.

The other major relationship that the TV show 10 Things and Shrew portray differently is Kate and Bianca, or Kat and Bianca.  In Shrew, Kate really has no interaction with Bianca, save to tie up her hands and try to make her confess which of her suitors she loves.  When their father comes in on this, he scolds Kate and comforts Bianca.  Kate replies:

...Nay, now I see

She is your treasure, she must have a husband....

Talk not to me, I will go weep,

Till I can find occasion for revenge. (II, i)

In 10 Things, these sisters have different personalities, but they ride back and forth to school together, and sometimes cover for each other when one disobeys the house rules.  Kat might not really understand why Bianca values the friends, etc, that she does, but their relationship is nowhere near as hateful as the one in Shrew appears to be.  Also, the above text suggests that Bianca is their father's favorite in Shrew, while in 10 Things, it seems that their father really admires Kat's quirky spirit best.