No. example.com and www.example.com are not the same site. You might choose to deliver the same content on both or redirect one to the other, but you don't have to do so - and Google doesn't assume that you have either.
Google treats example.com and www.example.com as separate entities. You should register both of them, whether or not one redirects to the other.
The crawl errors report for the non-canonical version is especially useful as it can show some incoming links that you might have otherwise overlooked and can recover. You'll also see separate crawl stats for each, but many of the other reports (such as HTML suggestions) will be blank for the non-canonical redirecting version.
On another note, Googlebot has been completely off the throttle for the last 24 hours. Hardly visited, after several days of being rather busy onsite.