In many questions about government, the terms "the state" and "the government" seem to be used almost interchangeably:
a common theme in the answer is that "the state" is a vehicle by which people agree to abide by standards of order as to how they interact with each other, and "the government" is the vehicle by which "the state" then enforces these agreements.
However, in real life, "the government" is actually two different entities, is it not?
a) "the government" as the agency that enforces agreements, as described above, but also (b) "the people who collectively work for the government," who often make sure that they themselves are taken care of before anyone else, and not infrequently, at the expense of everyone else.
We see Congress, for example, exempt itself from laws it imposes on everyone else.
We see state employees receiving large pensions (far larger than anyone in the private sector receives) even as states run large budget deficits and/or raise taxes on non-state employees to fund...
My perception is that distinctions of the sort you describe can be found but that they are both largley modern and contextual. So, one might determine the distinction in Hegel, Rawls, Foucault, etc. rather than find a uniform distinction across texts. A quick search of JSTOR raises this article that seems to offer some historical contextualization: “Theories of the Origin of the State in Classical Political Philosophy” by Harry Elmer Barnes, in a journal called “The Monist.” Vol. 34, No. 1 (January, 1924), pp. 15-62. Dwight Waldo’s book, The Administrative State (1948 but reissued in 2017 by Routledge), is a classic and makes an interesting distinction between the administrative and welfare state that may be helpful to you. As for the importance of the distinction, I leave that to others with more expertise in political philosophy, but my perception is that it is not terribly central. You will find some discussion of elite theory among political scientists. Trotsky’s critique of the Stalinist USSR...