Monitor mode: Sniffing the packets in the air without connecting (associating) with any access point.
Think of it like listening to people's conversations while you walk down the street.
Promiscuous mode: Sniffing the packets after connecting to an access point. This is possible because the wireless-enabled devices send the data in the air but only "mark" them to be processed by the intended receiver. They cannot send the packets and make sure they only reach a specific device, unlike with switched LANs.
Think of it like joining a group of people in a conversation, but at the same time being able to hear when someone says "Hey, Mike, I have a new laptop". Even though you're not Mike, and that sentence was intended to be heard by Mike, but you're still able to hear it.