How to do differential backups in a SQL Server RDS instance
From https://aws.amazon.com/premiumsupport/knowledge-center/native-backup-rds-sql-server/ 


  To set up a native backup of the SQL Server database, use the
  following services:
  
  An Amazon S3 bucket to store your backup files
  An AWS Identity and Access Management (IAM) role to access the bucket
  The SQLSERVER_BACKUP_RESTORE option added to an option group on the DB
  instance
  
  exec msdb.dbo.rds_backup_database  @source_db_name='database_name',
  @s3_arn_to_backup_to='arn:aws:s3:::bucket_name/file_name_and_extension',
  @overwrite_S3_backup_file=1;


After that you can use similar command to create diff backup:

exec msdb.dbo.rds_backup_database 
@source_db_name='database_name', @s3_arn_to_backup_to='arn:aws:s3:::bucket_name/diff_file_name_and_extension', 
@overwrite_S3_backup_file=1,
@type = 'DIFFERENTIAL';