Ping may be blocked by the firewall of your hosting company. Therefore ping requests can fail even though the URL requests work.
On the other hand, if ping would have worked, it is still not a good indication that your site is working. Your Apache/ISS server program may have crashed, or the server may be under severe load, sending 503 HTTP errors to many of the visitors. The only way to check this is a real HTTP checker which tests if the returned page is complete, and has the right status code (200).