There is a "transfer via ftp" function that will not create a backup archive, it just transfers all the files to website B. But you are still going to have to run the installation script to get the new database connection parameters, restore the database, update the configuration.php file, etc. These are just necessary parts of moving to a different server.
Dale L. Brackin
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!
My time zone is EST (UTC -5) (click here to see my current time in Philadelphia, PA)