Probably because it’s actually a MySQL problem that’s been highlighted because the new version of PHP is expecting the passwords to be stored in MySQL with a more secure hash. Reverting to an earlier version of PDO allows the currently stored password hash to function (16 bit), moving to the newer version of PDO breaks it (41 bit).
This is one of those things where you’ll have to educate yourself to be your own DBA and then get your hosting provider to help you fix the problem.
http://www.php.net/manual/en/migration53.incompatible.php
http://stackoverflow.com/questions/14612551/mysql-remote-connection-fails-with-unknown-authentication-method
http://dba.stackexchange.com/questions/34024/server-requested-authentication-method-unknown-to-the-client-mysql-old-password
How to troubleshoot the hashed password problem.
http://stackoverflow.com/questions/1340488/mysql-php-incompatibility/1340538#1340538