I'm a native of Chicago, where the "two-flat" is the analog to a "duplex." Unlike the arrangement of a duplex, in which the two apartments are on either side of a center wall, the apartments in a 2-flat are stacked, one above the other. Duplexes are common, but in southern United States, a duplex is sometimes called a "double-barrel" or a "shotgun double" and consists of two "shotguns" slapped together. Chicago 2-flats can be brick, stone or wood and almost always have a basement, which provides space for mechanical equipment, a laundry, and abundant storage space. Because Chicago has alleys, almost every 2-flat has a rear yard and a 2-car garage. I don't recall seeing garages in the Bayou St. John neighborhood of New Orleans, although many have driveways, and the duplexes sit over crawlspaces rather than above basements. Consequently, storage and laundry must be contained within the apartments. 

I think that it's safe to say that historically both styles accommodated working class families. In Chicago, it was typical for the owner to occupy the second floor apartment and to have a tenant on the first floor. Similarly, the owner of a duplex or double-barrel would occupy one side of the two units. The theory was that the tenant would help pay off the mortgage during the working portion of the owner's life and then provide an income stream once the owner retired. It was a good strategy and ultimately, the off-spring of the owners benefited once their parents passed on. 

Today, thanks to gentrification, lower incomes, and income inequality, many of these two family buildings are being converted into commodious, single family homes. 

These photos document duplexes in the Bayou St. John neighborhood of New Orleans, Louisiana.