I see this in this way:
By definiton, OutOfRangeException should be use to when potential problem is logical. This (OutOfBoundsException) is for runtime, so it's perfect for catching errors which occur because of bad result from database and simillar.

Example of using OutOfBoundSException (see also note in "OutOfRangeException class" article):

<?php
class HandleApplication {
    public function __construct($_POST) {
        if(!isset($_POST['secretCode'])
            throw new OutOfBoundsException('Application hasn't sent secret code for authenticate');
}

An example where an OutOfBoundsException can occur:
Lets say post a certain division process, we wish to access a value in an Array [provided ofcourse if the result value of the division is within the size of the Array]..

try{
    if ($iNum2 == 0){
        throw new Exception("Division by Zero");
    }
    $iResult = $iNum1 / $iNum2;
    echo ("Division result is: ".($iResult)."<br/>");
}
catch (Exception $e){
    echo ("Division by Zero is not possible.".($e)."<br/>");
}

$rg_Array = array(1,2,3,4);

try{
    if ($iResult > sizeof($rg_Array)- 1){
        throw new Exception("Exceeding key values");
    }
echo ("Capturing value from \$rg_Array post Division process:".($rg_Array[$iResult])."<br/>");
}
catch (Exception $e){
    echo ("Value of Division result is out of bounds for the array.".($e)."<br/>");
}
?>

OutOfRangeException is for Integers out of range.
OutOfBoundsException is for key values, not found in the target array.

Editor's note: This is incorrect; OutOfRangeException has *nothing* to do with Integer ranges. I decided to keep this highly voted comment even though it is wrong for education's sake.

I would probably repeat @jacub's answer, but I see the distinction as following:

`OutOfBoundsException` you throw, when _you_ _set_ bounds and do not wish them to be crossed by an external entity, but they may be.
Example: you ask for a number in range [1, 6] and user input 0 or 7 or what else. You throw an exception indicating user error they could amend.

`OutOfRangeException` you throw, when you expect value within allowed boundary, but actual operation fails.
Example: You are trying to access external API call using information provided by the API itself, and it suddenly tell you that the reference key is not pointing to a valid object. You throw an exception indicating internal failure that the user is unable to correct.

In both cases, the key word is "you throw". Both exceptions are generated at runtime, but the latter describes a situation that should not happen, ever, if a system is configured correctly.

class MyDynamicTastyPie implements ArrayAccess{
    private $_pointeur = 0;
    private $_array = ['strawberry slice','white chocolate','nuts'];
    
    public function offsetExists($key){
        return isset($this->_array[$key]);
    }
    
    public function offsetGet($key){
        if ($key > count($this->_array)){
            throw new OutOfBoundsException('Your tasty pie doesn\'t contain so slices');
        }
        return $this->_array[$key];
    }
    
    public function offsetSet($key, $value){
        $this->_array[$key] =$value;
    }
    
    public function offsetUnset($key){
        unset($this->_array[$key]);
    }
    
    public function addSlice($slice){
        $this->_array[] = $slice;
    }
}

try {
    $myDynamicTastyPie = new MyDynamicTastyPie();
    $myDynamicTastyPie->addSlice('Black chocolate cream decoration');
    echo $myDynamicTastyPie[7];
}
catch(OutOfBoundsException $e){
    echo 'Here is your OutOfBoundsException!';
}