This is part of the complexity of the Tax Code, but the grantmaker that rejected your application was just wrong.  Section 501(a) provides that organizations described under sections 501(c), 501(d), and 401(a) are exempt from federal income tax.  Section 501(c) now has 29 separate sections (See Ready Reference Page: “What Do We Mean When We Say ‘Nonprofit’?”), including 501(c)(3) which describes charities.  Section 501(d) describes certain religious and apostolic organizations, and section 401(a) describes qualified pension, profit-sharing and stock bonus plans.  None of the organizations that meet the definitions set out in these sections has to pay federal income taxes.

Section 501(c)(3) charities are further subdivided under Section 509(a), basically between public charities and private foundations, such as the Gates Foundation or the Ford Foundation.  Under the law, a charity is deemed to be a private foundation unless it can show that it is a public charity under section 509(a) of the Code.  Section 509(a) has four subdivisions: 

Subsection (a)(1) includes churches, schools, hospitals, and other charities that are publicly supported by a broad range of donors, including those described in section 170(b)(1)(a)(vi) as your organization is. 

Subsection (a)(2) includes charities that are broadly publicly supported primarily through fees for services, like a nursing home.  (See Ready Reference Pages: “Calculating Public Support Percentage” and “New Schedule A Reflects Change in Public Support Rules”)

Subsection (a)(3) describes “supporting organizations.”  (See Ready Reference Page:  “Supporting Organizations Qualify as Public Charities”) 

Subsection (a)(4) describes organizations operated for testing for public safety.  All of the organizations described in Section 509(a) are considered public charities.

Deductibility of contributions is controlled by Section 170 and does not apply to all 501(c) organizations.  Gifts to 501(c)(4) social welfare organizations or (c)(7) social clubs, for example, are not generally deductible.  Your being described in Sections 501(c)(3) and 170(b)(1)(A)(vi) means that gifts to your organization are deductible.  Your organization is a 501(a) exempt organization, a 501(c)(3) charitable organization, and a 509(a)(1) public charity.