One reason why Act 3, Scene 3 is important to the rest of the action of the play is that it foreshadows Romeo's upcoming suicide by giving us a chance to see just how youthfully minded and hyper-emotional Romeo is. Not only that, we see that Romeo's youth, irrationality, and hyper-emotionalism are actually the catalysts for his upcoming death.
In this scene, Romeo begins to moan and wail because he has been banished from Verona where Juliet is, saying,
'Tis torture, and not mercy. Heaven is here,
Where Juliet lives; and every cat and dog
And little mouse, every unworthy thing,
Live here in heaven and may look on her;
But Romeo may not. (III.iii.30-34)
In other words, Romeo is moaning because, as a banished person, he will no longer be able to look at Juliet.
However, Friar Laurence calls Romeo's tears "womanish," says that he "shamest" his "shape," meaning the shape of a man, and proclaims that he should rejoice in the fact that he has only been banished and is not dead (116-150). He further argues that so long as Romeo is alive they can soon proclaim to their families of his marriage to Juliet, make up with Juliet's family for Tybalt's death, beg the Prince for forgiveness, and then be allowed to return to Verona (157-158). In other words, Friar Laurence is pointing out what Romeo does not see, that so long as he is still alive he still has hope that this mess will turn out right.
The reason why Romeo's outlook in this scene is so shallow is that he is so young. All he can think of is being separated from Juliet, which makes him want to kill himself; he cannot see prospects in the future and see that things may turn out well in the end. This same youthful, shallow, and rash thinking leads him to commit suicide when he believes that Juliet has died, even though she does not even look dead when he sees her, proving that Scene 3 in Act 3 is important, because it shows us his youthful, shallow, rash mind, which are elements that lead to his upcoming death.