I am wondering if I should add the www prefix when I add a website to Google Webmaster Central. Both versions of the website (http://www.website.com and http://website.com) are exactly the same so I am wondering if it makes a difference.
It's really up to you, some likes websites with www because it makes the site look professional, and some likes without www because its shorter.
You can only use one though. Set a redirect on your choice. If you chose with www, set a redirect that redirects http://example.com to http://www.example.com and vice versa.
"www" is actually a subdomain. I you add example.com, you covered all subdomains, eg blog.example.com. If you verify www.example.com, you need to add blog.example.com later, not sure about WT, this is generally this way.
Choose one and only one version so search engines wouldn't be confused if your site is with www or not. I personally like without www but it really makes no difference.
Then make a redirect adding this code to .htaccess
<IfModule mod_rewrite.c>
RewriteEngine on
RewriteCond %{HTTP_HOST} ^www\.yoursite\.com$ [NC]
RewriteRule ^.*$ http://yoursite.com%{REQUEST_URI} [R=301,L]
</IfModule>
Then mark it in Google Webmaster tools.
It's not very good when half your backlinks go to www.yoursite.com and half of backlinks to yoursite.com