- Instead of shareholders, nonprofits have members. Members don't own the corporation (no one does) but control it insofar that they hire-and-fire its directors.
- Memberships normally can't be sold, and nonprofits have no equivalent to dividends.
- There are stronger conflict-of-interest protections, preventing officers from signing cheques to themselves or otherwise using the corporation's resources for their personal benefit - at least without someone else's approval.
A typical setup for nonprofits is to tie memberships to being a director, so that current board members choose their own successors. This is known as a "self-perpetuating board".
Regional differences may be relevant here: in Canada, you can incorporate a nonprofit corporation and later apply for charity status (which restricts the company's possible activities in exchange for a 0% income tax rate) but it's completely optional.
In the US, on the other hand, to the best of my knowledge, one has to establish a for-profit entity first and then apply to the IRS to recognize it as a charity (which is a lengthy process). The words "charity" and "nonprofit" are interchangeable in the US for that reason.
If the concept of a non-charitable nonprofit exists in your jurisdiction, you can get the benefits of a nonprofit structure (ie. no owners) without restricting the company's activities, minus tax exemption and being able to issue tax receipts.
For context: I founded a non-charitable nonprofit in Canada; I'm not super familiar with the US side of things so I'm happy to be corrected.
The US works similarly. You incorporate as a nonprofit, but you have to separately apply for tax-exempt status afterwards. If you don't, you're a nonprofit without the benefits of being tax-exempt.
These things are hard or maybe impossible in a non-profit (IANAL/correct me if I'm wrong)
Non-profits can make surplus revenues over costs and retain them; they can't return profits to shareholders or other particular beneficiaries.
Certain classes of nonprofits are restricted from certain business activities, or limited to certain activities. E.g., charities (501c3 nonprofits) must be organized and operated exclusively for purposes on an list of charitable purposes.