Since both terms are in everyday use, it makes sense to ask what is the difference between a nonprofit and a not-for-profit.
In a technical sense, it's probably correct to say there is no significant difference between the two terms. State "nonprofit" or "not-for-profit" corporation statutes sometimes use both terms side by side to suggest they are synonymous.
The IRS does make one distinction though. In some publications, the IRS explains that for them "not-for-profit" refers to an activity, for example, a hobby (like fishing). In contrast, "nonprofit" refers to an organization established for purposes other than profit-making. Note: nonprofit does not necessarily mean "charitable."
For example, a "nonprofit" organization can be an association of people who engage in the hobby of fishing – in other words, do it "not-for-profit." That matters, because you can't deduct expenses incurred in pursuing a not-for-profit activity. (A corollary is that if a business goes on not making any profit too long, the IRS may find that it's "not for profit" and deny deductions of related business expenses.)
More generally, though, the fact that there are two terms in use does not cause problems since everyone involved knows what everyone else means.
Caution: If you're talking about a possible job with a nonprofit, it's probably a good idea to avoid calling it a "business" or "firm" unless you're very sure that's what the people who work there already say. "Agency," "group," or "organization" are the words most often heard when nonprofit employees talk about the places they work.
There are lots of other terms that get used to refer to the broad realm of activities that happen voluntarily – without government edicts or demands to turn a profit. Collective nouns like these:
There are probably many more ways to communicate this idea. As long as you're being understood, you can choose whichever term fits your audience and suits your purposes.