PHPMailer SMTP Error Could not connect to SMTP host

Question

I am having trouble sending mail through PHP but it works when I send mail through Thunderbird.

This is my thunderbird settings:

Server name: mail.exampleserver.com

Port: 587

Username: user@exampleserver.com

Secure Authentication: No

Connection Security: STARTTLS

This is my PHP code:

 $mail = new PHPMailer();

 $mail->IsSMTP(); // send via SMTP

 $mail->Host = SMTP_HOST; // SMTP servers

 $mail->Port = SMTP_PORT; // SMTP servers

 $mail->SMTPAuth = true; // turn on SMTP authentication

 $mail->Username = SMTP_USER; // SMTP username

 $mail->Password = SMTP_PASSWORD; // SMTP password

 $mail->From = MAIL_SYSTEM;

 $mail->FromName = MAIL_SYSTEM_NAME;

 $mail->AddAddress($aSecuredGetRequest['email']);

 $mail->IsHTML(true); // send as HTML

What am I doing wrong here? How can I solve this?