WWW vs Non-WWW
Back in the early days of the internet, it was more than just the "world wide web". Usenet news, Gopher, FTP and IRC all played a large role in day-to-day lives of many internet users. It would be common for one domain to have "subdomains" for each of these, and it wasn't yet obvious that the web would become the defacto default. You would have ftp.example.com, irc.example.com, and www.example.com, and those may not even be on the same server. Today, [most of] those still exist and are used, but it's the "web" that made it big time.
So, today, when we think of example.com, we just assume it's the same thing as www.example.com, even if that isn't really guaranteed. It just so happens that the world standardized on it.
However, as far as some search engines go, they are not necessarily the same. Just like www.example1.com is different from www.example2.com, it can be that example.com is considered different from www.example.com. This issue falls under a larger category called canonicalization - the idea that out of many copies of a document, which is the original.
Google, for example, has a webmaster tools setting to set the canonical domain to one or the other. An even better approach is to use a 301 redirect to enforce it for the entire internet.
So, WWW or Non-WWW? It's a surprisingly polarizing issue. Personally, I always pick the WWW version, if for no other reason than it makes it very obvious when printed that you're talking about a website. I can imaging a portion of the population who wouldn't know what 1234.co means, for example, but would immediately understand www.1234.co. The key is in making sure that both of them work.