The difference between a monologue and a soliloquy is that a monologue is a long speech delivered by one character "who forgets or neglects" that other characters are still present while a soliloquy is delivered when the character is completely alone or at least believes himself/herself to be alone (Dictionary.com). Both monologues and soliloquies express the inner-most thoughts and feelings of the character. Soliloquies can also give the audience vital information (Dr. Wheeler, "Literary Terms").
One excellent example of a soliloquy is Juliet's opening speech in Act 3, Scene 2 in which she pours out her heart about waiting for Romeo and her feelings of anticipation for her wedding night. When Juliet says this long speech, she is alone in her room. Also, the speech shows us all her inner-most thoughts and feelings and characterizes her as growing into a woman; therefore, it is an excellent example of a soliloquy.
An excellent example of a monologue can also be found in Act 3, Scene 2. In lines 102-132, Juliet speaks to try and console herself about Romeo having just killed Tybalt. Even though she addresses Nurse in the first lines, reprimanding Nurse for thinking poorly of Romeo, she says to herself, "Shall I speak ill of him that is my husband?" Also, Juliet loses herself in this speech and talks as if she has forgotten that Nurse is still in the room. The speech becomes a means for her to grapple with her own feelings in order to help her realize that Romeo killed Tybalt because Tybalt was about to kill Romeo. The speech also serves to help her mourn the news that Romeo has been banished, which feels as bad as if all of her family has been slain, as we see in her lines:
'Romeo is banished'--to speak that word
Is father, mother, Tybalt, Romeo, Juliet,
All slain, all dead. (I.ii.127-129)
Hence, even though Juliet addresses Nurse at the beginning of the speech and again at the end, because this speech mostly addresses herself as a means of pouring out her feelings, Juliet speaks as though she has forgotten that Nurse is still present in the room. Therefore, this speech is an excellent example of a monologue.
Similar to a soliloquy, an aside is spoken by one character and it is believed that the other characters do not hear him/her. The differences are that an aside is very short, usually only consisting of a few lines, and also that an aside is directed straight to the audience. In addition, asides are always marked by stage directions while soliloquies are not (Dr. Wheeler, "Literary Terms"). One excellent example of an aside in the third act is spoken by Juliet. When Juliet's mother tells Juliet that she is weeping not so much over Tybalt's death but because Romeo, the "villain," is still alive and has been granted such a soft sentence by the Prince, Juliet addresses the audience with an aside to declare that Romeo is not a villain and that she has forgiven him, as we see in her lines:
Villain and he be many miles asunder.
God pardon him! I do, with all my heart;
And yet no man like he doth grieve my heart. (III.v.84-86)
Hence, we see the differences between monologues, soliloquies, and asides and have identified one of each in Act 3.