[fidgewidge]

"While there may be differences between libertarianism and anarcho-capitalism, I've struggled to see them in practice"

I don't think either are actually implemented in practice at the moment, although you can argue that certain aspects of certain politicians or their agendas lean libertarian, and that some societies have in the past been a lot more libertarian than they are today.

The primary difference is this: anarchists of any kind, ancap or not, see no role for the state whatsoever. Libertarianism requires a state to exist, but has a very clearly defined set of roles for it such that anything outside those roles is considered more properly the role of the private sector. At a minimum, the state is expected to implement:

- [Border] defense (armies, navies, passports)

- A violence monopoly within their territory (police, courts, jails, laws against murder, abuse, etc). Yes, this doesn't always sit easy with the US specific gun culture, but the "out" is to say you can own a gun only for self defense, hunting and overthrow of an out of control state, which doesn't infringe on this principle.

- Contract law (civil law courts, ability to levy financial penalties)

- Property rights (land registers, stock markets, bond markets and other financial infrastructure, also quite often IP rights)

- Sufficient taxation to fund those things and subsequent state functions

- Anti-monopoly enforcement

And a bunch of other things that are less well agreed on, but in some interpretations might involve attempts at limiting externalities.

So a libertarian state still has a relatively extensive civil service, it would still have a parliament or congress, tax authorities and so on, but it's smaller and more tightly focused than a non-libertarian state. It is actually implementable in the real world, today, without violating any of the known basics of human nature or government. Libertarians accept that some current roles of the state have evolved for sound reasons but propose that those roles can be done equally well or better by the private sector (e.g. instead of bank bailouts you have narrow banks). We know such countries can exist because they have also existed in the past, for example the early US government was much closer to this ideal than the current one.

Ancaps on the other hand propose a theoretical society that has no government at all yet is also peaceful and prosperous. No such society has ever existed, not even anything close to it, and there are many obvious open questions with no known answers. It's very different.