Usually if I talk about minarchism I'm going to essentially bash it and promote anarchism against it. This is to be expected, since I am an anarchist. But I would like to point out a certain sense in which I think that the conflict between anarchists and minarchists may be at least somewhat of a misnomer or even a false dichotomy. In particular, I think that there is a sense in which anarchists are de facto governmentalists and at least some minarchists are more or less closet anarchists or anarchists in denial.
To be clear, I'm more specifically referring to libertarian anarchism and libertarian minarchism, which is to say that both of them essentially share the same basic libertarian premises with regard to interpersonal relations such as freedom of association and individual sovereignty. And by minarchism I do not refer to any old vaguely "small government" philosophy, but specifically to what I would call "radical minarchism" or a strictly "bare bones" view of government. I am approaching this from the assumption that both libertarian anarchism and libertarian minarchism share the same basic premises.
Assuming that both more or less have the same underlying premises, the basic difference that distinguishes them can be thought of simply in terms of what conclusions are reached from those premises. At a basic level, libertarian minarchism proposes that the initiation of the use of force is wrong and concludes that we should have a government that is limited to the point at which it does not initiate the use of force, while libertarian anarchism proposes that the initiation of the use of force is wrong and concludes that we should have no government.
So here we arrive at the basic conflict between "limited government" and "no government". At this point, an interesting question that arises is the extent to which the disagreement between these two ultimate conclusions revolve around nothing more than semantics over the word "government". The minarchist tends to define "government" in a way that leaves open the possibility of having a government that does not initiate force, while the anarchist tends to define "government" as inherently involving the initiation of force.
While the minarchist proposes the ideal of a government that does not initiate force, often the anarchist responds to this by claiming that it wouldn't be a government then if it doesn't initiate force. Hence, if the minarchist truly is consistently opposed to the initiation of force (and this includes the tricky and radical part of opposing taxation and coercive barriers to competition), they are in some sense de facto anarchists and their "government" is little more than a homeowners association or some sort of voluntary mutual protection agency. This is part of the cognitive dissonance that tends to turn minarchists into anarchists.
Others prefer to make a formal distinction between "state" and "government". I have never personally been particularly comfortable with this distinction because I see a certain risk of "the state" merely being snuck in through the back door under the label "government". Nonetheless, if we are to make such a distinction, this seems to make the minarchist vs. anarchist conflict even more semantic in nature, to the point at which some "minarchists" may in fact be advocating a form of "government" that would be acceptable to an anarchist and some "anarchists" may in fact be advocating a form of "government" or at least advocating an idea that is compatible in theory with certain forms of "government".
Indeed, anarchists (except perhaps at the fringes of primitivism) tend to clarify that they are not opposed to social organisation itself and they favour a basic standard of justice. If the term "government" is conceived of as merely referring to any sort of social organisation and basic reasonable rules for interpersonal relations, then anarchists are de facto governmentalists precisely because they are not anti-social-organisation-in-itself. In backing away from that position, and probably with a lot of annoyance at constantly being misrepresented via cultural stereotypes, anarchists inevitably are forced to make a distinction between voluntary social organisation and "the state" as they understand it.
And so therein lies the interesting rub: a radical minarchist tends to advocate a "government" or "state" as defined specifically in terms of voluntary social organisation (at which point, from the anarchist's perspective, it is no "state" at all), while an anarchist tends to advocate a "stateless society" in terms of voluntary social organisation (and it ultimately makes no practical difference whether or not you slap the word "government" on to it). So it seems to be the case that if voluntary social organisation in general is what the common goal is, then there may very well be little to no meaningful difference between these positions beyond personal semantics.
Of course, by no means do I intend to argue that all minarchists are closet anarchists by definition. Quite frankly, in my judgement the vast majority of minarchists significantly fall short of consistently favouring voluntary social organisation, partially because the way things have been traditionally done is often taken for granted and people easily get sucked into reformism. So while libertarian minarchists may have a proto-anarchist political philosophy, in practise they often tow a more moderate line in which they defend the existing reality of "government" — which is to say a "government" that initiates the use of force in some way. Either the minarchist is blinded to the force or pragmatically endorses some level of it.
There are many minarchists who make glaring exceptions to their principles that are big no-no's from an anarchist perspective. For example, libertarian anarchists are opposed to taxation, while many minarchists (despite reoccurring quasi-anarchistic vocal opposition to the concept of taxation) acquiesce to the need for at least some limited form and rate of taxation (Rand was an exception to this, but she still clung to a doctrine of violent retribution theory and supported violence used to crush competitors of her "objectivist government", hence falling short of anarchism; but screw her, despite her influence she wasn't exactly a "libertarian" anyway).
But in all honesty, some of the exact same problems that tend to plague minarchism plague certain individuals and segments within the anarchist movement as well. Let's be crude about this: at least *some* self-proclaimed anarchists are either closet statists or implicitly statist without realising it, and this problem can be found in all segments of the anarchist movement ranging from anarcho-communism to anarcho-capitalism. Anarchists do face a certain danger of merely taking the form of social organisation that they previously have been biased towards and renaming it something else or merely sticking the word "voluntary" next to it or merely proposing a more localised version of the exact same thing. Certain self-proclaimed anarchists are, at best, minarchists that are a bit more radical than the rest.
It also works the other way around: at least *some* self-proclaimed minarchists are essentially advocating anarchism and are probably accused of being anarchists a lot (hell, I was accused of being an anarchist by people when I was a minarchist, and I used to back down from the accusation while still flirting with anarchism but not fully embracing it). When one is proclaiming that "taxation is theft" and referencing Lysander Spooner to reject the authority of the constitution while simultaneously clinging to minarchism, one probably has some cognitive dissonance to resolve. It makes sense why so many libertarian anarchists used to be minarchists; they resolved their cognitive dissonance, which pushed them into anarchism.
While I am most certainly biased (and for good reason, not mere prejudice) towards anarchism, looking back at the evolution of where my head is at makes me see some senses in which the distinction between anarchism and minarchism may not be as wide as a black and white analysis may imply and I can say that "I used to be in that position". I've found that anarchy, as a practical matter, is in some sense merely a different paradigm of "governance". Furthermore, the internal conflicts among anarchists helps illuminate the fact that anarchism is in some sense very concerned with social organisation, since a lot of the conflicts revolve around the compatibility of certain forms of social organisation with libertarian principles.
I have not set out to prove that anarchism and minarchism in general are necessarily indistinguishable, but merely to provide some food for thought that perhaps there may be a certain point at which the lines blur a bit, at least depending on the kind of minarchism that one encounters. Sometimes you might scratch a minarchist and find an anarchist inside, and sometimes you might scratch an anarchist and find a rather extreme authoritarian inside (*cough* curse the Hoppe cult!). The details of political ideas and the relationship between ideas in political philosophy can be rather complex sometimes. Perhaps we should be more sensitive to subtleties.