What is in a name?
Nonprofit, not-for-profit, tax exempt, or charitable organizations... many times people use these phrases interchangeably for organizations that have "legal and ethical restrictions on the distribution of profits to owners or shareholders" (wiki), in contrast to commercial enterprises. However, these terms can mean different things to different people.
• Nonprofit or not-for-profit
According to Foundation Center, the simplest way to distinguish between the two terms is to think of the term "not-for-profit" as an activity, such as a reading a book. The term "nonprofit" refers to an organization that is not intended to make a profit, say for instance, an adult literacy group.
When the two terms were used interchangeably, people may still have different reasons to prefer one over the other. I like "not-for-profit" as it is more literally true in the sense that these organizations can actually have profits (therefore not "non"profit), just that they are not "for" the purpose of "profit." That said, I actually use "nonprofit" (or even NPO) more often simply because it is more abbreviate.
• Nonprofit or (federal) tax-exempt status
According to IRS, non-profit status is a state law concept. Although most federal tax-exempt organizations are non-profit organizations (in their states), organizing as a non-profit organization at the state level does not automatically grant the organization federal (income) tax-exempt status. To qualify as exempt from federal income taxes, an organization must meet requirements set forth in the Internal Revenue Code. See Types of Tax-Exempt Organizations or Publication 557 for more information.
• Nonprofit, charitable, and public charities
Charitable organization (also known as a charity) is an organization with exclusive purposes of charity, that is, the giving of help to those in need. Charities are all non-profit organizations, however, not all non-profit organizations are charities. Nonetheless, the distinction may be drawn differently from common-sense understanding of "charitable" activities or from the legal definition of charitable status. In common-sense, a nonprofit association can be formed for people of a same hobby, for instance, fishing. Its activities are certainly not-for-profit but clearly far from charitable.
In legal terms, however, IRS calls all organizations under 501(c)(3) as charitable organizations, which are then separated into two categories: "private foundation and "public charities." Private foundations are defined in the Internal Revenue Code under section 509(a) as 501(c)(3) organizations which do not qualify as public charities, because private foundations are subject to more limitations in revenue sources as well as expenditure types. For instance, private foundations receives most of its income from investments and endowments (rather than grants or service fees). This income is used to make grants to other organizations, rather than being disbursed directly for charitable activities.