The Willow Song is significant because it is one of the few moments in the play where Desdemona shows her vulnerability.
We know that Iago is a villain, but we don’t know him well. He’s been a soldier all his life and he hasn’t had time to develop much of a personality outside of his job. So when he says things like “There’s no trust/But apparent and inconstancy rules” (Act 1, Scene 3) or “The Moor already changes with my poison” (Act 2, Scene 1), we don’t really understand what he means by this. But when Desdemona sings her Willow Song, she shows us that she understands what he means—she knows that there’s something wrong with Othello and she knows that Iago is responsible for it. We can see from her face how upset she is about what Iago has done to Othello, even though she doesn’t know why it happened yet. She has no idea how much danger she’s in because of her innocence and willingness to trust people unconditionally.
The Willow Song is significant because it foreshadows Desdemona’s death.
In Ancient Greek mythology, willows were associated with mourning.