SQL Developer's work is to write sql queries for differnt processes/applications. It includes writing stored procedures, triggers, functions etc. You'll need to learn the sql syntaxes to write the queries. However, more important thing is you'll need to identify the best ways to do any task.
Database Admistrator's work include administering the sql server. Activities performed by a SQL DBA is long but here are few of them:
1. Establishing sound backup policy for SQL Server databases
2. Ensuring the security of the databases & other securables
3. Configuring HA features for the databases such as clustering, mirroring, log shipping or replication etc.
4. Monitoring the perfomrance of sql servers & tuning the servers/queries to get the optimum performence.