This scene also firmly establishes the conflict of the play and also establishes the hopefulness for the future of Verona.  The over-arching conflict of Capulet vs. Montague is specifically stated here, and their conclusion is "so what?"  "What does that have to do with us?"  The audience is excited to see if this couple is going to be able to end the long-standing feud.  If the audience knows the play is a tragedy from the start, then they also know that Romeo and Juliet are going to likely end up dead and that the audience is now invested in watching how that tragic conclusion is going to resolve the bigger conflict, or not.

The balcony scene is where both Romeo and Juliet declare their feeling for each other. We know they have fallen in love when they first met, but this is the point where the characters become aware of each others feelings. Without this scene, the play could not progress in the same manner. This is the point where they decide what course their lives will follow. This scene leads directly to their marriage, their grief, and their deaths. It is a key in building the rest of the plot.

The balcony scene is pivotal to the plot as this is the scene in which Romeo and Juliet commit themselves to each other.  In addition, the poetic significance of the play is here established as the light/dark dichotomy becomes pronounced:

....Therefore pardon me,

And not impute this yielding to light love,

Which the dark night hath so discovered.

Daytime becomes dangerous for the two lovers, and night is introduced as the time of safety for their love.

The significance is that the budding attraction between Romeo and Juliet has a chance to blossom under the moonlight. The story would not exist if it were not for the balcony scene or some other very similar scene: they needed a place to meet alone and impress each other with their eloquence and passion in a culture that was very protective of contact between unmarried youths.

I think that this scene also shows us just how passionately in love Romeo is.  In a sense, it foreshadows his doom.  In this scene, we have Romeo knowingly facing death in order to see Juliet.  He is so in love that he does not care if he risks death.  This can be seen as a foreshadowing of how he will once again face death for Juliet's sake later in the play.

The significance of the balcony scene in Shakespeare's play Romeo and Juliet is mainly to illustrate an important tragic flaw the lovers possess, which is impulsiveness. A tragic flaw is a personality weakness that leads to the death or destruction of a character. This scene shows how young and impulsive Romeo and Juliet are and how this impulsiveness leads to their untimely deaths. The moment Romeo and Juliet meet to the time they perish all occurs within five days. That's it. In the balcony scene, they have met only moments before at Lord Capulet's party. They meet up again after the party in Juliet's garden and immediately profess their love for each other and decide to marry. They have just met and have already decided to marry the next day. This scene shows just how young and impulsive the two young lovers are.