I have worked for both for-profit and not-for-profit health systems. They both have to make a profit, or they can't stay in business. The difference is that in a for-profit system, the profit goes to the shareholders, whether publicly traded or private held shares. In a not-for-profit system the profit is put back into the business, and may be used for many things, one of which is executive compensation. Look at the not-for-profit entity's IRS Form 990. It will list the highest paid employees compensation, both regular salary and other compensation, i.e. bonus. It may shock you as to how much they are paying executives.
The other big difference, which most people don't realize, is a not-for-profit system doesn't pay any taxes, whether it be sales tax, property tax, or income tax. That accounts for additional profit that the for-profit system doesn't benefit from. They consume city services but do not financially support those services with property tax or sales tax.
The for-profit system gets to write off charity care as a business expense on their tax return. Since the not-for-profit system doesn't pay income tax, there may not be as big an emphasis on writing off bills. Some not-for profit systems are worse at hounding people for collections than for-profit systems.
An example of not-for-profit executive compensation is the Baylor Scott & White Healthcare system in Dallas, from their most recent Form 990. And this is just their first 10 listings. Yet they are not-for-profit. Three of the listings are former employees, still receiving over $1,000,000 a year.
Baylor Scott & White Health, Full Filing - Nonprofit Explorer - ProPublica