Does Google treat a website differently with www. and without www?

i think, there is no difference between this two cases!

Yes, that doesn't matter with the search engines.. what matters is the keywords that you use and its relevancy to your contents..

[FONT=Georgia]According to you Google will consider these two URLs as same website. If this is the case, I think it not have an impact on the PR of that website. Is it desirable to run a website with www and without www.[/FONT]

Google doesn't treat these URLs differently. It does count them as the same one.

yes, google treats both domains different.
even we can also see different domains on SERP on different rankings.

yes it depends upon what type of link you use while getting a back link in most of the cases www is not included in the search engines .

[FONT=Georgia]Thanks for the reply. Your replies help me find a solution to this problem. [/FONT]

I suggest that you use 301 redirection..