The Haori is a lightweight coat that was traditionally worn several ways by different groups of people.  Even today is an essential feature of the ceremonial kimono attire for men, along with the Hakama.  The Haori was not worn by women until the Edo period (1615-1868) when geisha started to wear the garment in imitation of men's kimono attire.  Women later wore the Haori mainly to prevent their kimono from becoming dirty or wet when out - it was worn as outerwear.  Originally, the Haori was worn exclusively by the upper classes, gradually becoming allowable for lower class wear.

The Hapi (or happi) appears to result specifically from the relaxation of social restrictions regarding marks of rank.  Considered a workers or artisan's jacket, the hapi holds  a place similar to the hippari (112 Japanese Field Clothing).  However, it is worn open, while the hippari is a folding type of kimono, closing with four ties.  

The Hapi is often decorated with emblems which have their origins in the the system of monsho - symbolized by the hereditary crests of Japanese families.  Historically, these crests were used to mark the formal kimono, household, utensils, lanterns, and gravestones of important families, as early as the Heian period (794-1185), and gradually became recognized as the household's special symbol.  As such, the crests indicated the position, honor, and taste of the family and could be quite intricate.

For military families of rank (daimyo), the use was very different.  On the battlefield, warriors needed to tell enemy from compatriot, and foot-soldiers needed a symbol with which to identify.  The earliest crests, worn on armor and banners, were of necessity simple, legible, and easily distinguishable.  

During the Edo period, the Hapi first appeared.  Servants customarily wore their master's crests.  The style of this work jacket (the Hapi) would have a large and bold decorative version of the crest dyed on the center of the back and on the front neckband.  

More history and lots of information on how to screen print a crest onto your handmade Hapi is in the 129 Hapi and Haori pattern. 

References: The Book of Kimono: Complete Guide to Style and Wear by Norio Yamanaka. Kodansha International. 1982.