OK, so let's look at the code piece by piece to see what is happening...
<?php
/* Open the votes.txt file and read the current number of votes. Load that into a variable ($votes) and close the file. */
$filename = 'votes.txt';
$fp = fopen($filename, "r");
$votes = fread($fp, filesize($filename));
fclose($fp);
/* Open the score.txt file and read the current score. Load that into a variable ($score) and close the file */
$filename = 'score.txt';
$fp = fopen($filename, "r");
$score = fread($fp, filesize($filename));
fclose($fp);
/* Create an average of the scores to votes using the above created variables and load that into a variable ($average) */
$average = $score/$votes;
/* Echo the $average variable to the user */
//echo("&averageText=Your vote changed the average of this movie to $average");
echo("&averageText=$average");
?>
There is absolutely nothing in this script that you have here that uses a variable from the URL, so the Register Globals setting wouldn't affect this script in any way.
Now, in the original script:
<?php
//this is the same code we used in average.php to get the values of votes and score
$filename = 'votes.txt';
$fp = fopen($filename, "r");
$votes = fread($fp, filesize($filename));
fclose($fp);
$filename = 'score.txt';
$fp = fopen($filename, "r");
$score = fread($fp, filesize($filename));
fclose($fp);
//Up votes by one and add the user's choice to the score
$votes++;
$score = $score + $choice;
$filename = 'votes.txt';
$fp = fopen($filename, "w");
fwrite($fp, "$votes");
fclose($fp);
$filename = 'score.txt';
$fp = fopen($filename, "w");
fwrite($fp, "$score");
fclose($fp);
?>
It's pretty much the same thing, except you reference a $choice variable that isn't accounted for in the script, so is that what is being passed by the variable? If so, then change this:
$score = $score + $choice;
to this:
$score = $score + $_GET['choice'];
and see if that works better for you.