Why www prefix is used with some websites and not with ...

I know it's not the best answer but here's wikipedia explanation for your question:

"WWW" is commonly found at the beginning of Web addresses because many organizations in the past followed a convention of naming hosts (servers) according to the services they provide. So for example, the host name for a Web server was often "www"; for an FTP server, "ftp"; and for a news server, "news" or "nntp" (after the news protocol NNTP). These host names then appeared as DNS subdomain names, as in "www.example.com".

This use of host or subdomain names and therefore such prefixes are not required by any technical standard; indeed, the first Web server was at "nxoc01.cern.ch" [2] and even today many Web sites are available without a "www" prefix. The "www" prefix has no meaning in the way the main website is shown; locally or around the world. The "www" prefix is simply one choice for a Web site's subdomain name.

Some Web browsers will automatically try adding "www." to the beginning, and possibly ".com" to the end, of typed URLs if no host is found without them. Internet Explorer and Mozilla Firefox will also prefix "http://www." and append ".com" to the address bar contents if the Control and Enter keys are pressed simultaneously. For example, entering "example" in the address bar and then pressing either just Enter or Control+Enter will usually resolve to "http://www.example.com", depending on the exact browser version and its settings.

I would like to add to Elton's answer, that www stands for "world wide web", so just as pop3.example.com or imap.example.com are incoming mail servers, smtp.example.com is example's outgoing mail server, www.example.com is example's web site.

Of course you can install a web server on any computer, whether or not it has an address that the DNS maps to a name that begins with www.

It is common practice to use www as a prefix, because that's what people expect. If you hear about a company named "ixia" and would like to find out about it, you'd type "www.ixia.com" in your browser. admins know this, and that's why they always use the www. domain.

The reason to consider using www. is because by nature people will type in www. or say go to www. yeddayeddayedda .com.

Bloggers and younger webmasters are removing the www. because it seems to be a trend. However the business world is not following suit and it is very common to see tv ads stating www. So what we have now is a mix of www.'s and non-www sites.

From a marketing perspective: Since TV advertisers and a percentage of users still use and say www. in branding websites, it may be wiser to follow the old-school and business world. Leaving off the www. is mainly seen in blogs. In reality adding a canonical redirect will resolve the issue. So the difference is not going to have any impact on marketing but it does make clear that it is a website, just as much as having the .com does.

From an SEO standpoint: It is wise to fix a potential canonical issue common in sites using non-www domains, as they do not properly redirect the www. to the non-www and this can cause a "canonical" (authoritative) path issue, which in turn, can cause ranking issues.

So whatever you determine to use whether www. or non-www, remember, that if you do not redirect the non-www to the www and someone links to your site with the opposite, it may cause a duplicate version of your site and thus a ranking issue may incur. It sounds dopey, it is, and it does happen, so addressing the issue will help. Fixes can be found in this 301 & Canonical Redirect Tutorial.

From a personal opinion: Having been involved in marketing since the 90's, I think domains look naked without the www. I prefer it and recommend its use because the majority, as I see it, still use www.