Depends on your priorities.
If performance is you absolute driving characteristic, then by all means use the fastest one.  Just make sure you have a full understanding of the differences before you make a choice
- JSON converts UTF-8 characters to unicode escape sequences. serialize()does not.
- JSON will have no memory of what the object's original class was (they are always restored as instances of stdClass).
- You can't leverage __sleep()and__wakeup()with JSON
- Only public properties are serialized with JSON
- JSON is more portable
And there's probably a few other differences I can't think of at the moment.
EDIT
A simple speed test to compare the two
<?php
ini_set( 'display_errors', 1 );
error_reporting( E_ALL );
//  Make a bit, honkin test array
//  You may need to adjust this depth to avoid memory limit errors
$testArray = fillArray( 0, 5 );
//  Time json encoding
$start = microtime( true );
json_encode( $testArray );
$jsonTime = microtime( true ) - $start;
echo "JSON encoded in $jsonTime seconds<br>";
//  Time serialization
$start = microtime( true );
serialize( $testArray );
$serializeTime = microtime( true ) - $start;
echo "PHP serialized in $serializeTime seconds<br>";
//  Compare them
if ( $jsonTime < $serializeTime )
{
    echo "json_encode() was roughly " . number_format( ($serializeTime / $jsonTime - 1 ) * 100, 2 ) . "% faster than serialize()";
}
else if ( $serializeTime < $jsonTime )
{
    echo "serialize() was roughly " . number_format( ($jsonTime / $serializeTime - 1 ) * 100, 2 ) . "% faster than json_encode()";
} else {
    echo 'Unpossible!';
}
function fillArray( $depth, $max )
{
    static $seed;
    if ( is_null( $seed ) )
    {
     $seed = array( 'a', 2, 'c', 4, 'e', 6, 'g', 8, 'i', 10 );
    }
    if ( $depth < $max )
    {
     $node = array();
     foreach ( $seed as $key )
     {
      $node[$key] = fillArray( $depth + 1, $max );
     }
     return $node;
    }
    return 'empty';
}