PHPMailer: SMTP Error: Could not connect to SMTP host

I've used PHPMailer on several projects but now I'm stuck. It gives me the error:
SMTP Error: Could not connect to SMTP host.
I've tried sending email from Thunderbird and it works ! But not through PHPMailer ... Here are the settings from Thunderbird:
Server name: mail.exampleserver.com
Port: 587
Username: user@exampleserver.com
Secure Authentication: No
Connection Security: STARTTLS
I've compared these with the server at my last project where I used PHPMailer and they were:
Server name: mail.exampleserver2.com
Port: 465
Username: user@exampleserver2.com
Secure Authentication: No
Connection Security: SSL/TLS
My php code is:
 $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
Where I am wrong?

Solution