Fastest way to copy table between databases

[Wim Roffel]

I have two databases and I want to copy a table from one to the other. My ideal code would look something like:

$oldconn = mysqli_connect(OLD_SERVER, OLD_USER, OLD_PASSWORD);
$newconn = mysqli_connect(NEW_SERVER, NEW_USER, NEW_PASSWORD);
$query = "INSERT INTO `newserver.`newdb`.`newtable` SELECT * FROM `oldserver.`olddb`.`oldtable`";
mysqli_query($newconn, $query);

But that doesn't even work when the servers are the same and only the database accounts differ.

So what is the fastest alternative? Export to file? Select/insert row by row?