There is an important difference between the monitor and mute button.
Whenever you enable monitor, the audio from that particular source (whether it's a song, an external microphone, or both) will come in through your headphones or speakers.
Whenever you enable the mute button, that particular source will be muted to your audience. So even if your playlist is playing, your audience won't be able to hear it if it's muted.
In other words, the monitor button controls what you hear, whereas the mute button controls what your audience hears.