I consider myself a left-libertarian. To avoid any confusion over what this may imply, I fully support private property, voluntary exchange, money, rent, employment, and so on (or more strictly speaking, I don't advocate their abolition). And I completely oppose the state. I advocate a free market in everything, from clothing and shelter to defense and arbitration. I have a dislike for people like Noam Chomsky, who I feel is largely economically illiterate and confused. I'm not a marxist or a communist or a syndicalist. Some may therefore be thinking, "so what's so 'left' about it? what differentiates you from 'right' libertarians? you sound like any other anarcho-capitalist to me!". I'd like to explain myself in order to make it clear that there is a very real distinction to be made.
Firstly, it is worth exploring how one views power in general. All libertarians, particularly market anarchists, oppose the power of the state. A lot of emphasis is placed on the power of the state and how it effects society. However, in my understanding, while the left-libertarian joins their comrades in opposing the state, they oppose the concentration of power and centralization in general. This includes the concentration or centralization of so-called "private power". While cookie-cutter anarcho-capitalists make brilliant arguments against state power, they tend to specialize so much in doing so that they may neglect the problems with the concentration of "private" power. Their libertarianism is "thin" in the sense that it is restricted to anti-statism.
The cookie-cutter anarcho-capitalist often seems to act as if whatever is "private" is legitimate in all respects. It's almost as if the principles somehow magically don't apply when we are dealing with non-state organizations. But to use a simple example, a gang or mafia may be "private" but it certainly is not legitimate. The left-libertarian views matters more broadly, that is, they apply libertarian principles not only to delegitimize the state but also to any other group of "private" people who violate rights. The left-libertarian's libertarianism is "thick" in the sense that it is more than just a matter of anti-statism, it is more broadly a matter of anti-authoritarianism and anti-centralization. The left-libertarian may additionally oppose corporations, extremely large buisinesses and possibly even organized religion. The left-libertarian sees no good reason why buisinesses should be centralized.
Karl Hess once described "the right" as supporting the concentration of power into the fewest hands possible, while in contrast "the left" stands for spreading it about as much as possible in an equilibrium. "The left" implies "equality of authority" in which everyone's freedom is limited by the like freedom of everyone else - a mere restatement of the non-aggression principle. Using this analysis, right-libertarians are to "the left" to the extent that they oppose the concentration of power in the hands of the state, but they nonetheless are still to "the right" to the extent that they still support private concentrations of power. While the right-libertarian may be consistantly anti-state, they are not consistantly opposed to the concentration of power. They may even fully endorse "private" concentrations of power and portray such organizations as victims of the state.
In short, the right-libertarian or cookie-cutter anarcho-capitalist, while they are likely fully aware and informed of the fact that we don't currently live in a free market or free society, functions as a "vulgar libertarian". What this means is that they function as apologists for big buisiness, corporations and currently existing conditions or property titles. They use free market theories or analysis to legitimize conditions and organizations that came about in a non-free market. They tend to cling to a worldview in which "big buisiness is America's most persecuted minority", as Ayn Rand once stated. They still tend to think of state intervention as somehow being inherently anti-buisiness, despite overwhelming evidence to the contrary. The right-libertarian is essentially pro-buisiness more or less across the board without proper consideration for context. The left-libertarian calls them out on this.
Another difference between the left-libertarian and the right-libertarian is over what they think society will be like in the absence of the state. Cookie-cutter anarcho-capitalists essentially envision a society more or less identifical to currently existing society but without the state. But the left-libertarian sees much more broad implications that would seem to radically alter the organizational structure of a society. The left-libertarian does not think that the results of a free market would mirror current economic conditions by any stretch of the imagination. Left-libertarians may tend to think that free competition would function as a check on the general size of economic organizations, and therefore draconian large buisinesses simply couldn't survive or exist. They may also be tolerant of or more open to possible "socialistic" experiments within a free market, or advocate a signficant increase in self-employment over standard wage-employment.
The difference between the two sides can also be thought of in terms of how one's position relates to the traditions of the anti-authoritarian left, or how one views their own position in relation to it. It's partially a matter of historical context and the political spectrum. Right-libertarians buy into the cliche that socialism is inherently a statist/political system, while left-libertarians aknowledge the existance and possibility of voluntary or anarchistic socialism (in short, all they're really doing is taking an anarchist without adjectives approach). To the right-libertarian, all socialist forms of organization are inherently violent or political systems - all socialism is state-socialism. To the left-libertarian, there is a distinction to be made between state-socialism and genuinely libertarian socialism. The left-libertarian has a much greater degree of tolerance for "socialistic" forms of organization so long as they are voluntary, while the right-libertarian considers all "socialistic" forms of organization to be inherently involuntary.
There's a major difference in terms of where one finds their roots. To the right-libertarian, their philosophy derives from and grew out of the "old right" and the founding fathers of America. To the left-libertarian, their philosophy derives from and grew out of the old libertarian left (the mutualists, the individualist anarchists, the voluntaryists, etc.) and wouldn't exist without them. The left-libertarian sees market anarchism as having grown out of old non-state socialist traditions and is likely to see ideas such as mutualism as not really being that far off from their own position in the grand scheme of things. In contrast, the right-libertarian is largely out of touch with such roots and probably considers mutualists and other more voluntaristic socialists to be enemies. They see little to no connection between these ideas and contemporary market anarchism, where the left-libertarian does.
Another major difference is over strategy and where one thinks their true alliances lie. The left-libertarian is much more likely to be opposed to the political process and consequentially they may not vote, argue against running for office and regularly denounce the libertarian party and reformism. The left-libertarian is a radical and a revolutionary. In contrast, the right-libertarian essentially functions as a minarchist in practise as they regularly participate in the political process, encourage people to participate in it, run for office themselves and advocate reformist strategies. Comparatively, the right-libertarian is a gradualist and even counter-revolutionary. The right-libertarian more or less takes the exact same strategy that a minarchist would, and consequentially falls prey to political oppurtunism and get-liberty-quick schemes.
The difference over where one thinks their alliances are is also significant. Right-libertarians regularly ally with conservatives, particularly paleoconservatives. To the right-libertarian, conservatism is the closest thing to libertarianism on the political spectrum and conservatives inherently are less statist then "the left". They may even views themselves as an extension of the conservative movement. The left-libertarian, in contrast, wants nothing to do with conservatism and sees no reason why it should be regaurded as somehow less statist than "the left". The left-libertarian sees conservatives as hijacking the libertarian movement and employing quasi-libertarian rhetoric to get people to associate their own positions with liberty and free markets. To the left-libertarian, conservatism in the original sense of the term is the polar opposite of liberty, as it stands for the status quo, the romantisization of the past and an endless sea of authoritarian tendencies.
From the perspective of the left-libertarian, sometimes the right-libertarian takes positions on current issues that in fact are conservative rather than libertarian. One of the most common cases of this is over the issue of immigration, in which right-libertarians essentially support restricting people from crossing political borders. To the left-libertarian, this merely grants legitimacy to the state and treats it as if it were a legitimate private property owner. The same is true of many so-called "privatization" schemes in which the state sells "its" property off to a single economic organization, essentially transfering from a state held monopoly to a private monopoly. The left-libertarian is much more skeptical of so-called "free market" reforms than the right-libertarian is, being much more likely to consider them manifestations of mercantalism or corporatism.
Another difference between the two may simply be a matter of cultural traits or preferances. Right-libertarians may often be strict "cultural conservatives" and therefore have traits such as opposition to multiculturalism, feminism and secularism. They may openly praise "the family", "the church" and "the nation". In contrast, the left-libertarian is much more likely to see these things such as multiculturalism and secularism as being good and support voluntaryist versions of them. The left-libertarian may add things such as anti-racism and anti-patriarchy to their agenda, and such things need not be imposed by the state but a result of voluntary efforts. And while many right-libertarians may tend to praise "the family", the left-libertarian may very well be skeptical about the organizational structure of many families and view them as abusive. And perhaps most importantly, the left-libertarian is not a nationalist.
It should be clear at this point what the left-libertarian is not: they are not vulgar libertarians, conservatives, in bed with conservatives, anti-immigrationists, reformists, extreme gradualists, and so on. It is likely (although not necessarily mandatory) that they are not racists, organized religion supporters, nationalists, chauvenists, and so on. The left-libertarian is not an apologist for "private" concentrations of power and corporations. The left-libertarian may very well oppose corporations. In short, the left-libertarian has distanced themselves from conservative traits as much as possible and view themselves as supporting liberty in a much more broad sense than your cookie-cutter anarcho-capitalist does. It is in the context of this much more broad perspective that they are to "the left" of their comrades.