What's the difference between fashion designing and fashion merchandising?

Fashion designing is actually designing the clothes. Fashion merchandising is the business side.

Noun 1.
Fashion designer - someone who designs clothing
Clothes designer, couturier, designer

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Mer·chan·dis·ing also mer·chan·diz·ing (mûrchn-dzng)
n.
The promotion of merchandise sales, as by coordinating production and marketing and developing advertising, display, and sales strategies.

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Most merchandisers will know something about fashion, but a designer need not know anything about merchandising if he/she has a manager to look after the business side.

Currently most who are in the fashion business will know a lot about merchandising too due to the high competition in this field.