Both terms are frequently used and frequently seen. There is no firmly established distinction, though some people have strong preferences between them.

"Not-for-profit" refers to an activity, for example, stamp collection or coin collection.

"Nonprofit" refers to an organization established for purposes other than profit-making. Note here that nonprofit does not necessarily mean "charitable."

Also note that the use of the term "non-profit" with the dash in between the letters. There is no implied or distinctive definition attached to this word (as far as I know). I have noticed the use of this term to introduce concepts
or explain technical aspects of organization when an audience is *not* familiar with nonprofit operations.

According to the IRS, "not-for-profit" refers to an activity, such as a hobby like restoring antique grandfather clocks in your home. "Nonprofit" refers to an organization established for purposes other than making a profit, such as the American Red Cross.