Use an SqlDataAdapter when wanting to populate an in-memory DataSet/DataTable from the database. You then have the flexibility to close/dispose off the connection, pass the datatable/set around in memory. You could then manipulate the data and persist it back into the DB using the data adapter, in conjunction with InsertCommand/UpdateCommand.
Use an SqlDataReader when wanting fast, low-memory footprint data access without the need for flexibility for e.g. passing the data around your business logic. This is more optimal for quick, low-memory usage retrieval of large data volumes as it doesn’t load all the data into memory all in one go - with the SqlDataAdapter approach, the DataSet/DataTable would be filled with all the data so if there’s a lot of rows & columns, that will require a lot of memory to hold.