If parent domain is www, does it matter if subdomain on a different server is non-www?

[SEOteamfl]

If you have a main website (www.example.com) with a subdomain of the website (service.example.com) that lives on a separate server with a separate IP address, is there an SEO benefit/advantage to have having the www included in the url since the parent url includes the www?

Assume:

1. Applicable 301 redirects are in place on both sites

2. No duplicate content issues

Additionally, would your answer be different if the site is a .gov or .edu site vs. a .com?

[FedeEinhorn]

Basically, www domains are a subdomain of example.com. You don't need to have your subdomains like www.service.example.com, if that was what you were asking. That's just poorly structured URLs IMHO.

There shouldn't be any different for edu or gov domains.