Matt Morris
A good question. It's not helpful that courts in California have frequently used the terms "public purpose" and "charitable purpose" interchangeably.
Taking your last question first: the IRS will not likely deny a 501(c)(3) determination if you check the box for "public purpose." There are existing 501(c)(3)s in California who told the state that they are formed for a "public purpose" and still obtained IRS exemption under 501(c)(3). The IRS cares about your description of your activities, and wants to see that you meet the definition of a charity in Section 501(c)(3): "organized and operated exclusively for religious, charitable, scientific, testing for public safety, literary, or educational purposes, or to foster national or international amateur sports competition ... or for the prevention of cruelty to children or animals."
Prior to 1978, the nonprofit statute in California only talked about charitable and religious purposes. You would think that if the legislature made a point of adding and distinguishing "public purpose" from "charitable purpose" in 1978, they must have had some good reason to draw a distinction. Not really. The legislature described the difference as this: "Public Benefit Corporations include but are not limited to the traditional 'charitable' corporations. ... [T]here is no need to define what is a 'public' or 'charitable' purpose. Instead, this question is left to those who form Public Benefit Corporations. If reasonable people could say that the purpose was 'public' or 'charitable,' the organization may be formed and operated as a Public Benefit Corporation. The State has little need to limit or define those purposes which are truly 'public' or 'charitable'...." That's not very helpful. That said, you could probably get lawyers to debate endlessly what might be a "public purpose" that is not strictly "charitable." One example that comes to mind is a nonprofit organization that is dedicated to preserving the architecture of an historic neighborhood. That seems like a stretch to call "charitable," but might be a public purpose. Or a nonprofit that exists to buy property to make walking trails in a wealthy neighborhood. Probably a "public purpose" but probably not "charitable."
Note that the statute requires you to describe your activities if you choose "public purpose." If you are planning to ask the IRS for Section 501(c)(3) status, I would expect that you can describe your programs in a way that falls into the IRS's requirement that the group be "exclusively for religious, charitable, scientific, testing for public safety, literary, or educational purposes, or to foster national or international amateur sports competition ... or for the prevention of cruelty to children or animals." If you talk to a lawyer who practices in the area of advising nonprofits, you should focus on talking about your planned activities, because that is far more important to the IRS than whether you tell California that you are for public purposes or charitable purposes.