It is different. Corporations can then go and list their holdings as a REIT on a Stock Exchange, thus allowing the general public to benefit from their arrangement. When private ownership is renting housing, you have either Banks (in the case of servicing debt) or individuals profiting at the expense of the general public.

Like the sibling commenter said, there are different venues for providing rental supply, sometimes making a public or private investment.

Again, I am not advocating either way as I am not in real state industry to have perspective on it.