You'll need to use the session handler that Joomla uses. Joomla saves session data to the database by default, but PHP uses files by default. So you're accessing the wrong medium for the data.
Here is how to load Joomla in a standalone app, which is probably the easiest way. The important part is the path to Joomla, which must be set in JPATH_BASE.
// Set flag that this is a parent file
define( '_JEXEC', 1 );
// replace this with your Joomla path
define('JPATH_BASE', '/path/to/joomla/');
define( 'DS', DIRECTORY_SEPARATOR );
require_once ( JPATH_BASE .DS.'includes'.DS.'defines.php' );
require_once ( JPATH_BASE .DS.'includes'.DS.'framework.php' );
$App =& JFactory::getApplication('site');
// now you can use Joomla functions
// session
$Session =& JFactory::getSession();
// $Session->set('example', 'remember me');
// $Session->get('example');
// or directly via
$_SESSION;
// $_SESSION['example'] = 'remember me';
// $_SESSION['example'];
// user - a good alternative (see JUser::setParam())
$User =& JFactory::getUser();
//$User->setParam('example', 'remember me');
//$User->getParam('example');
You can also just load the session files in libraries/joomla/session/ if you don't want the overhead of the whole Joomla framework (about 3 Mb). A bit more involved.
The other thing is the domain must be the same so cookies can be read. Even
www.example.com is not the same as example.com in this case.
You can set the session cookies to be read across sub domains, but I would not recommend it as it is not a secure practice (It opens XSS from any subdomain). See:
http://www.php.net/manual/en/session.configuration.php#ini.session.cookie-domain