In Ubuntu I normally install both but what are the differences between the client and server for MySQL.
As a bonus, when a new statement mentions that it needs MySQL 5.
The mysql server package will install the mysql database server which you can interact with using a mysql client. You can use the mysql client to send commands to any mysql server; on a remote computer or your own.
The mysql server is used to persist the data and provide a query interface for it (SQL). The mysql clients purpose is to allow you to use that query interface.
The client package also comes with utilities that allows you to easily backup/restore data and administer the server.