Found the answer on Reddit:

"Listening node" means that a full node has a specific port open (usually 8333), which allows for inbound connections and allows other peers to download older blocks from it (when that other node is syncing to the network, f.ex).

"Non-listening node" means port 8333 is closed and that node doesn't allow other nodes to download old blocks from it.

Both listening and non-listening nodes are full nodes (as long as they verify everything from scratch).