A subdomain (aka add-on domain) would be new.mywebsite.com. A subdirectory (aka subfolder) would be mywebsite.com/new. For purposes of SEO (search engine optimization), Google treats a subdomain as a separate entity.
So if mywebsite.com is just doing great – lots of page rank, backlinks, authority, etc., none of that will be of any help to new.mywebsite.com.
Subdirectories will benefit when other parts of the same domain are doing well; so that mywebsite.com can “pass through” some of its page rank and authority to mywebsite.com/new. This is one of those things that SEO types like to debate over – so be prepared for different opinions on this.
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