Check whats your need, if you are building up much bigger apps go for a fully fledged framework, rest for small apps you can use microrouters , according to your need, each has its own unique features.
One of the benefits of using a url router is access control - you could have a problem with just letting every file be executable by url, especially if you happen to have a remote file inclusion issue anywhere. With a router you know explicitly what can get invoked and what can't.
Although, doing it without a router is probably always going to be faster.
A late trick I discovered, was that you can configure your php instance to inclued a file with every script. For example the following can be added to .htaccess:
php_value auto_prepend_file 'before.php'
You can then use that for bootstrapping, front controller and to limit access. Can be a bit clumsy, but it's nicely divorced from simple page controllers.
Edit: as in, which router do you pick and how do you pick one? Choice can lead to decision paralysis.