Try changing it to:
$email = "me@me.com"
$subject = "Hi";
$message = "Hi again";
$name = "John Smith";
$fromEmail = "me@me.com";
mail($email, $subject, $message,
"To: ".$name." <".$email."> \r\n" .
"From: ".$fromEmail."\r\n" .
"Reply-To: ".$fromEmail."\r\n" .
"X-Mailer: PHP/" . phpversion());
Taken from the PHP site:
Note: Make sure you do not have any newline characters in the to or subject, or the mail may not be sent properly.
/* additional headers */
$headers .= "To: Mary <mary@example.com>, Kelly <kelly@example.com>\r\n";