$db = new SQLite3('/directory/file.db');
That's the SQLite3 extension, see here: http://php.net/SQLite3.
$db = new PDO('sqlite:/directory/file.sqlite');
That's the PDO extension with the sqlite driver, also for SQLite3 databases. See http://php.net/manual/en/ref.pdo-sqlite.php.
If you like PDO's interface, this is nice. Otherwise, SQLite3
is fine too. See https://stackoverflow.com/a/10703665/476 for what the difference between PDO and other extensions are.
$db = new SQLite('/directory/file.db');
This is:
In PHP 5.1, the SQLite extension also provides a driver for SQLite 2
databases; while it is not technically a part of the PDO_SQLITE
driver, it behaves similarly, so it is documented alongside it. The
SQLite 2 driver for PDO is provided primarily to make it easier to
import legacy SQLite 2 database files into an application that uses
the faster, more efficient SQLite 3 driver. As a result, the SQLite 2
driver is not as feature-rich as the SQLite 3 driver.