<?php
$data = $likeArray->getArrayCopy();
?>
will NOT be magically called if you cast to array. Although I've expected it.
<?php
$nothing = (array)$likeArray;
?>
Here, $data != $nothing.

If you did something like this to make your constructor multidimensional capable you will have some trouble using getArrayCopy to get a plain array straight out of the method:
<?php
public function __construct( $array = array(), $flags = 2 )
{
    // let’s give the objects the right and not the inherited name
    $class = get_class($this);

    foreach($array as $offset => $value)
        $this->offsetSet($offset, is_array($value) ? new $class($value) : $value);

    $this->setFlags($flags); 
}
?>

That’s the way I solved it:

<?php
public function getArray($recursion = false) 
{
    // just in case the object might be multidimensional
    if ( $this === true)
        return $this->getArrayCopy();

    return array_map( function($item){
        return is_object($item) ? $item->getArray(true) : $item;
    }, $this->getArrayCopy() );
}
?>

Hope this was useful!

Is there a difference between casting to an array and using this function?

For instance, if we have:
$arrayObject = new ArrayObject([1, 2, 3]);

Is there a difference between these:
$array = (array) $arrayObject;
vs
$array = $arrayObject->getArrayCopy();

If not, is there any scenario where they would produce different results, or do they produce the result in different ways?

"When the ArrayObject refers to an object an array of the public properties of that object will be returned."

This description does not seem to be right:

<?php
class A
{
    public $var = 'var';
    protected $foo = 'foo';
    private $bar = 'bar';
}

$o = new ArrayObject(new A());
var_dump($o->getArrayCopy());

/*
Dumps:

array(3) {
  ["var"]=>
  string(3) "var"
  ["*foo"]=>
  string(3) "foo"
  ["Abar"]=>
  string(3) "bar"
}
*/
?>

So it does not only include the public properties.

When I used print_r ($fruitsArrayObject) instead of print_r ($copy), i.e. ignoring the getArrayCopy() step, I still got the same output. Why?