Listening and non-listening nodes do exactly the same thing. They still send and receive blocks and transactions from their peers and will relay blocks and transactions to their peers.
Do non-listening nodes change their 8 peers more often than listening nodes?
Connections, both inbound and outbound, are rarely closed unless one peer goes offline. So listening and non-listening nodes will change their peers at about the same frequency. Outbound connections in general might change more frequently as we have stricter criteria for who we connect to and when to disconnect from them than we do with inbound peers.