Perhaps you'll get a better answer from a Unix user, but I'll provide what I know.
Your server needs a thread that opens a 'listening' socket that waits for incoming connections. This thread can be the main thread for simplicity, but can be an alternate thread if you are concerned about UI interaction, for example (in Windows, this would be a concern, not sure about Unix). It sounds like you are at least this far.
When the 'listening' socket accepts a connection, you get a 'connected' socket that is connected to the 'client' socket. You would pass this 'connected' socket to a new thread that manages the reading from and writing to the 'connected' socket. Thus, one change I would suggest is managing the 'connected' socket in a single thread, not two separate threads (one for reading, one for writing) as you have done. Reading and writing against the same socket can be accomplished using the select()
system call, as shown here.
When a new client connects, your 'listening' socket will provide a new 'connected' socket, which you will hand off to another thread. At this point, you have two threads - one that is managing the first connection and one that is managing the second connection. As far as the sockets are concerned, there is no distinction between the clients. You simply have two open connections, one to each of your two clients.
At this point, the question becomes what does it mean to "service them differently". If the clients are expected to interact with the server in unique ways, then this has to be determined somehow. The interactions could be determined based on the 'client' socket's IP address, which you can query, but this seems arbitrary and is subject to network changes. It could also be based on the initial block of data received from the 'client' socket which indicates the type of interaction required. In this case, the thread that is managing the 'connected' socket could read the socket for the expected type of interaction and then hand the socket off to a class object that manages that interaction type.
I hope this helps.