Whichever you pick, you should use that one and redirect the other to it, just so there is one consistent set of URLs.
nasa.gov doesn't resolve because www.nasa.gov is a CNAME to Akamai hosts for load balancing purposes, so they can't use the same thing for just nasa.gov (and manually inserting A records would get out of date and not distribute load properly). Google has the same setup (www is a CNAME), but because they control all of the hosts that the CNAME resolves to, they can put those same host records as google.com.
A couple of years a go I would have said to keep the "www" - it's a subdomain that refers specifically to the World Wide Web (or HTTP) address of a site, as opposed to the FTP or email services.
Nowadays, I prefer to either drop or ignore the "www" - if you're using an HTTP client to reach my domain, I'm going to go ahead and assume you want the website and not the FTP site. Intelligently named subdomains are good for organization :)
So either make both work, or redirect one to the other. It really doesn't matter very much either way.
The reason is that any cookies you set for the root domain (without www) will be sent on each request to any subdomain. So if you want to server images from images.domain.com, you'll get all the login, tracking, etc. cookies sent along with each request. 99% of the time, this is wasted bytes.
You could argue that the widespread use of CDNs for static content makes this point irrelevant, and perhaps you're right.
Simple as that