You may need to contact your host about it. It looks like the mysqli driver cannot perform a SHOW TABLES command against the database to list its contents. Since the old mysql driver works that rules out the possibility of wrong database credentials (username / password) or inadequate database permissions. Therefore the only reasonable explanation is that the PHP mysqli driver is broken or compiled against the wrong version of the MySQL client library. This is something that only your host can help with. In the meantime you can still use the old driver.
Please note that the code we use internally to connect to the different PHP drivers (such as MySQL, MySQLi and PDOMySQL) has remain unchanged for almost a year. Moreover, it's that code that we use ourselves to backup our non-Joomla! sites with Akeeba Solo. That's how I have ruled out a bug in our code: the code is unchanged and we've witnessed it working on several different environments besides our dev/test machines and servers.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!