1. Processing power, memory, and other system resources can be dedicated exclusively to your databases, which often take up more resources than other services.
2. It will add another layer of security by having your data on a separate machine from your web server. If someone finds a way to exploit the web server through a script, they still may not be able to access your databases.
3. You can run backups and other resource-intense activities without directly affecting the performance of the web server.
4. You can run the two servers from different locations and still have the same functionality.
5. You can add more database servers or web servers independently, depending on your disk space vs. bandwidth requirements.
Some of the disadvantages are that you will now have to run updates and maintenance on two machines rather than one. Your database server will still need an underlying operating system that you must keep up-to-date and running. Furthermore, your bandwidth and energy costs will likely increase. If you lease the servers, you will have to pay for an extra one. If these issues are not a big concern of yours, however, choosing to run your databases on a separate server might be a good idea.