Does not work for iOS yet? Edit: "At this time, WhatsApp Web is available only for Android, Windows Phone, Nokia S60, BlackBerry and BB10 smartphones." https://www.whatsapp.com/faq/en/web/28080003
Apple's browser is probably lacking necessary features. They have been much slower adopting features than the competition.
It's not a matter of speed. This seems to be using WebRTC for handling communications between the phone app and the browser. Safari has chosen not to implement WebRTC and so far it is still not a proper standard so I can see why. Google, as the developer of WebRTC, will obviously support it right away. Note that no browser other than Chrome can be used as well.