Thematically, Emilia and Bianca are important to the play because they act as foils for Desdemona and her relationship with Othello.  Emilia is the worldly, cynical wife of Iago who thinks "Men are stomachs" who belch women when they are through with them.  Her ethics are relative, and she would cheat on her husband if the price were right.  Desdemona, however, is unconditional in her love for Othello; and quickly excuses his bad temper as a result of his work. She would not cheat on her husband "for the whole world."   

Bianca is Cassio's courtesan.  Even though Bianca and Cassio are obviously fond of each other, their relationship will never lead to marriage.  They are of two different classes.  Cassio's laughter at Iago's teasing him about Bianca does much to show the disrespect that courtiers can have for women of that type. 

Men's views of women form the heart of the play.  Cassio uses Bianca for sex and jokes about their relationship.  Iago uses his wife as a subserviant who should obey him when he tells her to be quiet or to steal a handkerchief.  Othello tends to put Desdemona on a pedestal. But when he thinks she is unfaithful, he treats her like a whore, with contempt and rage. 

In exploring these relationships, Shakespeare shows how fragile and complicated the ties are between men and women. 

Emilia and Bianca are so important because they play critical roles in Iago's plan.  Iago exploits Emilia's relationship with Desdemona to get his hands on Desdemona's handkerchief.  He tells Emilia to steal it for him, yet he does not tell her why he wants it.  Emilia is loyal to her husband, so she takes the handkerchief without question.  Her unknowing help thus plays a critical role in Iago's scheme.

Bianca is also unknowingly used in Iago's scheme.  Iago puts Othello in a position to hear a conversation between Iago and Cassio, and although the two are talking about Bianca, Othello believes that they are talking about Desdemona.  Bianca has fallen madly in love with Cassio, but he does not return a similar level of affection.  So he speaks of her with some disinterest, allowing Iago to give Othello the impression that Cassio feels this way about Desdemona.  For these reasons, Emilia and Bianca are critical characters in the play.

Why are Emilia and Bianca so important in the play Othello?

In additions to the points mstultz72 has made, it is important to consider their functions in the plot or story of the play.

Both do contrast with Desdemona:  Emilia's less than ideal relationship with her husband, Iago, contrasts strongly with the deep passion between Othello and Desdemona; while Bianca is spoken about by Cassio as a mere plaything and certainly not wife material in any sense.

But they also serve the plot of the play as characters in their own rights, especially as concerns the event surrounding the handkerchief.

Emilia finds the handkerchief that Desdemona drops and, instead of returning it to her mistress, she gives it to Iago.  She then lies to Desdemona,...

Why are Emilia and Bianca so important in the play Othello?