I have designed a website before 5 years using PHP MySQL and still works fine without any issues. I heard MySQL is officially deprecated as of PHP 5.5 and has been removed a
Lets take a look at both of these extensions.
PDO
PDO is database neutral - You only need to learn one API to work with dozens of databases
So if you decide to move to another database, the only thing you would be changing is the DSN(data source name).
Support of both name and '?' placeholders for prepared statements.
Drupal uses PDO.
Mysqli
On the other hand Mysqli is designed specifically for Mysql databases and is recommended by Mysql. Mysqli works with Mysql and MariaDB databases.
Support of only '?' placeholders for prepared statements.
Joomla uses Mysqli
Conclusion
There are many conflicting arguments weather Mysqli or PDO runs faster. Both Mysqli and PDO use the same underlying drivers to access the database making the performance not worth comparing.
So there is no clear winner when working with Mysqli or PDO. But let's say you use Mysqli and then later you want to use another database, it would be a difficult transition.
The main strength of PDO over Mysqli is name placeholders for prepared statements and which is why I chose PDO over Mysqli.