Good question. It is true that not-for-profits and the government have a lot in common. Most importantly, they're both supposed to work directly for the public good (rather than for their own profits). There are some important differences, though:

1) Not-for-profits do not have monopoly on violence. (Most of the below follow from that.) 2) People do not have the same sense of entitlement visavi not-for-profits 3) You normally can create more of a "community spirit" within a not-for-profit. In governmental organizations, users have more of a tendency to view the leadership as "parents" to whom one goes to request more stuff. People take more responsibility in not-for-profits, especially small ones (this is closely related to 2)). 4) We might want to have a dispersed distribution of power in society. If so, we should not give too much power to the government, but would be better off giving power to other entities, such as not-for-profits.

Not-for-profit organizations working either for the public good or for some interest group such as workers, farmers, etc, have always had lots of influence in democratic societies. Often they've taken or been assigned tasks that the government could have done, such as health care, education (e.g. in the case of churches), unemployment insurance (unions), etc. They have of course always been legally regulated, and some of them have been closely related to political parties (e.g. churches - Christian democrats, unions - social democrats). However, the distinction between non-governmental organizations and governmental organizations has, by and large, been upheld in democratic countries (though it wasn't in, e.g. Sovjet Union).