I don't think so. Let's say you have one of the new TLDs for your site, say "new-york.florist", a nice domain. It's my belief that writing "new-work.florist" on your ads, business cards or merchandise will not let regular users know that it's a domain name. Writing www.new-york.florist will help indicate that this is your website.
Adding all the new TLDs have made the www prefix more useful than ever.
Sweet, unless you use any other subdomains for web pages, or want to in the future.
Usually you don't want cookies for domain.com to affect api.donain.com, admin.domain.com, test.domain.com etc.
I actually love www subdomains - the origin tells you it is a website.