Also, the comparison between the two is often confused by the fact that people tend to think of politics as a single left/right line. This is mostly because we as citizens vote for representatives, instead of voting for policies directly. And since we only get to vote for one representative, their different policies all get lumped together. In reality, the political spectrum is more like a 5D matrix, where things like personal freedom/authoritarianism, fiscal policy, religious freedom, etc exist on entirely separate axes.

One of the big differences between anarchists and libertarians tends to be fiscal policy and corporate regulation. Anarchists still tend to want things like public utilities, roads, trash collection, public art, universal healthcare, etc… They tend to see these as acceptable forms of government. Anarchists tend to be about collective action, and these things are an extension of that. They’re things that are too big to realistically build on an individual level, and they benefit everyone.

The “anarchy” part really comes into play when discussing personal freedoms, as anarchists tend to rebel against restrictions on what they are allowed to do. They tend to argue that individuals should have a lot more personal freedom, and local society should be correcting bad behaviors through social pressure (and use of force, if it comes to that). Break the social contract, and you’re punished by your neighbors until the behavior is corrected.

Anarchists also tend to argue for heavy corporate regulation, because monolithic “too big to fail” corporations will be able to unfairly exert external pressure on local communities. An example of this in action is Walmart running a new store at a net loss until all of the locally owned grocery stores are priced out and forced to close, at which point Walmart is the only grocer and can increase their prices. It’s not a perfect example, because anarchists tend to be against private property in general. Meaning the “locally owned grocers” would be more like a collective neighborhood garden. But it at least gets the point across.

To make a bad metaphor: An anarchist believes the government should pay for the neighborhood’s road using taxes, but the local neighborhood gets to decide what the speed limit (and other various rules of the road) should be.

All of those public works I listed in the second paragraph are things that libertarians would prefer to remove from the government’s purview entirely, by saying that a private company should be able to take over them instead of using taxes to pay for them. Libertarians are definitely more in the “every man is an island” basket. They tend to see public services, utilities, etc as frivolous government overreach. They tend to think that people should pay private companies to do these things, instead of paying taxes to have the government do them.

This stems from the idea that a private company will be more efficient than the government, which would conceivably lower costs while improving quality and agility. If you don’t like how a company does something, you can use the free market to find (or create) a new company instead. Essentially, under libertarianism, you’re not beholden to whatever the government decides to do, and libertarians think that extreme personal freedom should extend to corporations as well. They tend to argue for market deregulation and fewer government programs as a result.

To extend that bad metaphor: A libertarian thinks each neighbor should maintain their own section of the road out of their own personal effort/funds, and each homeowner gets to decide the rules for using their section of the road.