This great article by Joleen Deatherage in Philanthropy Journal really hits the mark on why branding is important for nonprofits.  She says:

"It is critical that nonprofits understand the importance of developing a strong nonprofit brand and how your brand literally lives within the minds of your audience.

Lack of clarity about or within an organization is usually the result of the absence of a core brand that connects with the intended audience.

I like to describe a brand as the "personality" of your organization, but it is much more than that. Your brand is the sum of all experiences at all touch points. It is the "gut feeling" someone has about your organization."