How do I set up apache so domains like example.com and www.example.com both serve the same content?

You should add a ServerAlias - modify your VirtualHost to look like this:

<VirtualHost *:80>
    # This first-listed virtual host is also the default for *:80
    ServerName www.example.com
    ServerAlias example.com 
    DocumentRoot "/www/domain"
</VirtualHost>

You can also have all subdomains redirect to your root

ServerAlias example.com *.example.com

As ceejayoz says in his comment, you need a ServerAlias

<VirtualHost *:80>
    ServerName www.example.com
    ServerAlias example.com
    DocumentRoot /var/www/public_html/www.example.com
</VirtualHost>

Whether you have the ServerName or the ServerAlias as www is really a matter of preference. It can make a difference if you subscribe to the no-www belief or the yes-www belief, as if you are using the UseCanonicalName directive, you might find Apache redirecting your users to somewhere slightly unexpected.

Also as ceejayoz mentioned, you want to choose a single camp (www.example.com or just naked example.com) and stick with it. You don't really want to be serving the same content on both variations - the alias'd one should really redirect to the canonical one. There are a thousand sites out there that will tell you how to set this up with mod_rewrite.

For Nginx:

  server {

  server_name  www.example.com;

  rewrite ^/(.\*) http://example.com/$1 permanent;

For Apache Example:

  <VirtualHost \*:80>

  ServerName  example.com

  ServerAlias www.example.com

</VirtualHost>

Restart Service after changes:

beside your apache question:
www.example.com should be a CNAME for example.com. One A record is enough.