Both monologues and soliloquies are long speeches delivered by a single character in a play; however, the difference between a monologue and a soliloquy is that a monologue is delivered in the presence of other characters while a soliloquy is delivered when the character is alone or believes he/she is alone. In contrast, a monologue is delivered when other characters are still on stage, but he/she forgets that the other characters are present.
One good example of a soliloquy can be found at the very end of the opening scene when Helena is left alone onstage (I.i.231-256). Helena is grieving over the fact that Demetrius is in love with her best friend Hermia when all throughout Athens Helena is recognized to be "as fair" as Hermia (232). Helena is also lamenting that the more Demetrius dotes on Hermia, the more she loves Demetrius (235-237). She further proclaims in this speech that Demetrius has forgotten the vow he made to marry Helena before he met Hermia (247-250). Finally, as a favor to Demetrius out of love for him, she announces her plan to tell Demetrius that Hermia plans to cross through the woods to elope with Lysander (251-256).
While there are many long speeches found in this play, most cannot be classified as monologues because the speaker never forgets that he/she has an audience. The long speeches are replies to what someone else has just said, granted, very long replies. However, there are a few things that characters say to themselves while choosing to ignore other characters present. One speech we could classify as a monologue, albeit a short monologue is spoken by Lysander while Hermia still lies sleeping and Helena flees his side after accusing him of mocking her by pretending to be in love with her. Lysander gives a short speech to himself describing how his feelings for Hermia have changed and how he now hates her while Hermia sleeps (II.ii.137-146). Since he is saying these things to himself as well as to the sleeping Hermia we can say that he has chosen to ignore his audience and that this speech can therefore count as a monologue.