mysqli was introduced about a year ago, before 3.0 was released: https://github.com/silverstripe/sapphire/commit/6bf3f7ded400c925dd66df2c2612c095d447c7b0#model/MySQLDatabase.php
2.4.x still uses the old mysql functions, it was upgraded during the course of 3.0 development because the majority of shared hosts we found had enabled mysqli by default. PHP have also suggested that everyone switches to mysqli or PDO MySQL instead of the old functions because they are not actively maintained any longer.
Excerpt about the old mysql API taken from http://nz.php.net/manual/en/intro.mysql.php:
"This extension is not recommended for writing new code. Instead, either the mysqli or PDO_MySQL extension should be used. See also the MySQL API Overview for further help while choosing a MySQL API."
I would suggest to your host that they enable mysqli, especially considering PHP themselves recommend everyone use it over the old driver. For the most part, I don't see that being a problem for them, because mysqli will run happily alongside the old mysql drivers.