Mail delivery is a tricky business... just because you send mail does not mean that anyone will receive it. Many receiving servers will simply ignore the incoming message if it does not meet certain criteria (in my experience Gmail and Hotmail are particularly prone to just silently denying delivery, so it doesn't even end up in SPAM). There are a few things to make sure you've done:
1) You've set up PTR/SPF (reverse lookup) entries in your DNS records
2) Make sure that you're not on any blacklists (http://www.mxtoolbox.com/blacklists.aspx)
3) Expand your headers
$headers = "MIME-Version: 1.0\r\n"
."Content-Type: $contentType; charset=utf-8\r\n"
."Content-Transfer-Encoding: 8bit\r\n"
."From: =?UTF-8?B?". base64_encode("Your sending display name") ."?= <$from>\r\n"
."Reply-To: $replyTo\r\n"
."X-Mailer: PHP/". phpversion();
However, if you really want to ensure that mail gets through, send mail via SMTP. You can never guarantee mail delivery, but it will be much more reliable. If you're not sending a large volume of mail, you might try using Mandrill or similar service to relay emails for you.