The Mysql-client package allows your to connect to a MySQL server. It'll give you the ' MySQL ' command-line program.
The Mysql-server package allows-to-run a MySQL server which can host multiple databases and process queries on those D Atabases. It'll give you the MySQL daemon, a.k.a Mysqld.
The "MySQL" package probably includes both of the above.
If you just need to connect to a remote server and run queries, install just mysql-client. If you need to host a database, install the client and server.
The MySQL server package would 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-use this query interface.
The client package also comes with utilities, allows you to easily backup/restore data and administer the server.
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Mysql-server/mysql-client differences