You can send the email and use the template by using the Mandrill PHP API wrapper.
require 'Mandrill.php';
$mandrill = new Mandrill('YOUR_API_KEY');
$message = array(
'subject' => 'My subject',
'from_email' => 'marc@example.com',
'to' => array(array('email' => 'recipient1@example.com', 'name' => 'Marc')),
'merge_vars' => array(array(
'rcpt' => 'recipient1@example.com',
'vars' =>
array(
array(
'name' => 'FIRSTNAME',
'content' => 'Recipient 1 first name'),
array(
'name' => 'LASTNAME',
'content' => 'Last name')
))));
$template_name = 'YOUR-TEMPLATE-NAME';
$template_content = array(
array(
'name' => 'main',
'content' => 'Hi *|FIRSTNAME|* *|LASTNAME|*, thanks for signing up.'),
array(
'name' => 'footer',
'content' => 'Copyright 2013.')
);
$response = $mandrill->messages->sendTemplate($template_name, $template_content, $message);
print_r($response);
If you want to use the SMTP via SwiFtMailer, you could call the Render API method to render a template, which will give you the full HTML, which you can pass to SwiftMailer but that seems a bit of a long winded way of doing it compared to the PHP API wrapper.