I'm going to speak as to how it applies in the United States.
There are scenarios where it is beneficial to be a non-profit, but other scenarios where it probably isn't and likely where it is beneficial it may not be for tax reasons. Such an example would be an organization that was seeking funds from sources which required it (some funding sources are charities or governments that give money away for certain causes, etc).
I'll give you some examples. There are different kinds of entities as far as taxes go and they each get taxed differently. These entities sometimes tax just the individual's income (Sole proprietor, LLC, partnerships, etc). That means that the entity itself doesn't pay any taxes, but its owners pay it on there own income taxes. Other entities like a sub chapter s corporation pay taxes on the profits at the end of year, and the shareholders pay a tax when they go to sell the stock. However because businesses are out to make grow they are probably re-investing that profit and thus not paying tax on it. However the employees (including owners) who then get a paycheck are going to have to pay income taxes like any other employee. I believe the reason people end up with a corporation is that the taxes on selling shares is less than the taxes on income and so if your salary is low you'll end up paying less in taxes.
Now if you look at a non-profit they might pay nothing in taxes, but the employees will still have to pay taxes. So there really isn't a benefit for tax purposes. And for people who don't know this taxes are not paid on good that are resold and any goods bought for the business get written off the taxable income. So ultimately the business can avoid taxes on that anyway.
Taxes are basically on the profits and there are different ways to structure things such that you pay less tax, but ultimately what it comes down to is the money you pull out of the non-profit, for-profit, etc ends up getting taxed in some that impacts the business owner at some level.
* I'm not an expert on this stuff, it's why ThinkPenguin, Inc has an accountant, etc, but this is my general understanding of it.
