var_dump() shows the information in your browser.
Mage::log() echos information to your system.log file (var/log/system.log).

To use Mage::log(), you need to enable logging in the backend: System > Configuration > Developer -> Log Settings -> Enabled > Set to "Yes".

Furthermore you can force logging using the forth parameter of Mage::log() set to true:
Mage::log($foo,null,'system.log',true);

According to the function in Mage.php:
public static function log($message, $level = null, $file = '', $forceLog = false)

Mage::log() will be especially useful if you do not see the output in your browser (debugging SOAP API requests, AJAX requests,...).