Make sure to choose one in Google Sitemaps (I am not yet used to the new name, webmaster central or sthing like that), then optimize for the one you chose. About which one you chose it doesn't actually matter but you will not want to have both appear in th SERPs.
I submit with www and also do 301 redirection. People have practice of using www. If I am going to link you I will use www.domain.com so its better to submit with www.
hiiiii yes, defenately google consider both as a different URL's with the www and without www this is known as the "canonical issue" of the URL's and you will get the backlinks for those for which you will do the submissions so the URL you want to make popular U should submit that URL