A client-server network model has client nodes requesting services from a server node which responds with a service. A peer to peer network model has nodes both requesting and serving services.
An example of a client-server model is when you go to a website on your laptop (client node) and the website (server) responds with the webpage.
An example of a peer to peer model is Bitcoin. The ledger where all the transactions are recorded is not in one central location but are contained across multiple nodes (peers).
The client-server model is centralised while the peer to peer model is decentralised.
The client-server model can be more stable as it’s supported and typically available 24./7 and in a server room situation. But it can be less stable due to the centralised nature – if the server goes down for any reason no clients can connect.
The peer to peer model can be more stable as the service can be distributed across multiple nodes. However, it can be less stable and secure due to no guarantees the necessary nodes will remain available.