For setting up the timeouts while making REST calls we should specify both these parameters but I'm not sure why both and exactly what different purpose they serve. Also, what if we set only one of them or both with different value?
CONNECT_TIMEOUT is the amount of time it will wait to establish the connection to the host. Once connected, READ_TIMEOUT is the amount of time allowed for the server to respond with all of the content in a give request.
How you set either one will depend on your requirements, but they can be different values. CONNECT_TIMEOUT should not require a large value, because it is only the time required to setup a socket connection with the server. 30 seconds should be ample time - frankly if it is not complete within 10 seconds it is too long, and the server is likely hosed, or at least overloaded.
READ_TIMEOUT - this could be longer, especially if you know that the action/resource you requested takes a long time to process. You might set this as high as 60 seconds, or even several minutes. Again, this depends on how critical it is that you wait for confirmation that the process completed, and you'll weigh this against how quickly your system needs to respond on its end. If your client times out while waiting for the process to complete, that doesn't necessarily mean that the process stopped, it may keep on running until it is finished on the server (or at least, until it reaches the server's timeout).
If these calls are directly driving an interface, then you may want much lower times, as your users may not have the patience for such a delay. If it is called in a background or batch process, then longer times may be acceptable. This is up to you.