New & Returning Visitors
A new visitor is a visitor that never visited your site before. Therefore, a visitor is new if current session is his first session on the site. On all pages visited in this current session the visitor will be considered new. On any subsequent sessions he is going to be considered a returning visitor.
A session lasts as much as the visitor is browsing the site with no more than 20 minutes between different page views.
Assume the date range is May 1-31.
User A visits www.domain.com for the 1st time on May 5th and then returns to the website on 12th and 17th of May.
User B visits www.domain.com on April 8th and then again on May 5th and 12th.
In Unique Visitors: User A will be counted as a First Time (New) Unique Visitor on May 5th, and User B will be counted as Prior (Returning) Unique Visitor. User A and User B will not be counted as a Unique Visitor (either first time or prior) for any subsequent visits (May 12 and 17th).
In New vs. Returning: The visit on May 5th from User A gets counted as a visit from a "New" visitor. Subsequent visits from User A (May 12 and 17th) get counted as visits from a "returning" visitor. All of User B's visits are counted as visits from a returning visitor.
To sum it up, Unique Visitor report counts visitors to your website (counting each visitor only once in the selected date range) whereas New vs. Returning classifies all visits (sessions) in the date range by the visitor type.