It used to be so that Google would index www and no-www separately. To fix that, people would usually use a 301 from one to the other to prevent duplicate content, or alternatively use the rel="canonical" tag to point out which had their preference.
Now that Google Webmaster Tools provides the option to select a preferred domain it doesn't matter much anymore in terms of Google SEO. By selecting a preference Google will use the one you chose and ignore the other. So no, for Google you won't have to use a 301 anymore, and there is no difference in SEO results between the two. Choose the one you prefer.
Keep in mind though that (althoug it's a small percentage) there's other search engines out there that may still see the www and no-www as different sites. It's therefor adviced to still keep either a 301 or a rel="canonical" to point them to the right one. If your aim is just Google, you won't really have to, but if you can just use one of the two just to be sure.