Depending on your website setup and the traffic volume that you are experiencing. The move to a dedicated database server should be considered if the traffic warrants it. Any sluggish performance could be addressed with a dedicated server that handles the database functions and nothing else.
If you are hosted on a dedicated server and have a lot of different things from email, ftp, http server, to database server on one machine. It may get overwhelming trying to stretch your limited resources on that one server to do everything. With the addition of a server that is dedicated to handling the database functions can relieve the CPU usage and memory resources being used on your primary server.
A dedicated database server would use all the CPU power and memory available on the dedicated database server to handle the database queries and nothing else. With only the database as the only service running on the server you would free up the existing CPU and memory on your primary server to be used by your http, mail, ftp, or firewall resources.
The dedicated database server would be able to handle sudden spikes in traffic that would result in a surge of read and writes to the database server than having your primary server hosting everything. The dedicated database server would also be easier to manage and maintain since all you have is the basic operating system and the database software running on the database server.