Here is Noam Chomsky discussing the difference between the two:
From an AFAQ:
However, due to the creation of the Libertarian Party in the USA, many people now consider the idea of "libertarian socialism" to be a contradiction in terms. Indeed, many "Libertarians" think anarchists are just attempting to associate the "anti-libertarian" ideas of "socialism" (as Libertarians conceive it) with Libertarian ideology in order to make those "socialist" ideas more "acceptable" -- in other words, trying to steal the "libertarian" label from its rightful possessors.
Nothing could be further from the truth. Anarchists have been using the term "libertarian" to describe themselves and their ideas since the 1850's. According to anarchist historian Max Nettlau, the revolutionary anarchist Joseph Dejacque published Le Libertaire, Journal du Mouvement Social in New York between 1858 and 1861 while the use of the term "libertarian communism" dates from November, 1880 when a French anarchist congress adopted it. [Max Nettlau, A Short History of Anarchism, p. 75 and p. 145] The use of the term "Libertarian" by anarchists became more popular from the 1890s onward after it was used in France in an attempt to get round anti-anarchist laws and to avoid the negative associations of the word "anarchy" in the popular mind (Sebastien Faure and Louise Michel published the paper Le Libertaire -- The Libertarian -- in France in 1895, for example). Since then, particularly outside America, it has always been associated with anarchist ideas and movements. Taking a more recent example, in the USA, anarchists organised "The Libertarian League" in July 1954, which had staunch anarcho-syndicalist principles and lasted until 1965. The US-based "Libertarian" Party, on the other hand has only existed since the early 1970's, well over 100 years after anarchists first used the term to describe their political ideas (and 90 years after the expression "libertarian communism" was first adopted). It is that party, not the anarchists, who have "stolen" the word. Later, in Section B, we will discuss why the idea of a "libertarian" capitalism (as desired by the Libertarian Party) is a contradiction in terms.
I said anarchism is an extreme form of libertarianism. I wasn't implying they are really different, just two extremes of the same thing
You are separating the two as if all anarchism is libertarianism, but not all libertarianism is anarchism. From the traditional, and historical perspective, libertarianism is completely synonymous with anarchism (as shown above), thus they are equal terms in the minds of those who originally used them.
The Libertarian Party of the US has taken this term, obviously, but I believe what most anarchists would believe--the use of 'libertarian' here is illegitimate since unfettered capitalism would necessarily impinge on the freedom of others. This is obvious just in the example of the wage slavery system that exists under state-protected capitalism today. Under unfettered capitalism, potentially, this could be even worse since private owners would have, presumably, no responsibility to the people.
As Mikhail Bakunin, an early anarchist, said "Liberty without socialism is privilege, injustice; socialism without liberty is slavery, brutality".
the latter repudiates the legitimacy of 'most forms of economically significant private property.'
Anarchists are opposed to private property, but not opposed to personal possession--and there is an important distinction between the two. To anarchists, and in Proudhon's words,
property is theft or more directly
"Property . . . violates equality by the rights of exclusion and increase, and freedom by despotism . . . [and has] perfect identity with robbery." [Proudhon, What is Property, p. 251]
From an AFAQ:
Firstly, [the property is theft maxim] recognises the fact that the earth and its resources, the common inheritance of all, have been monopolised by a few. Secondly, it argues that, as a consequence of this, those who own property exploit those who do not. This is because those who do not own have to pay or sell their labour to those who do own in order to get access to the resources they need to live and work (such as workplaces, machinery, land, credit, housing, products under patents, and such like).
In other words, private property is the state writ small, with the property owner acting as the "sovereign lord" over their property, and so the absolute king of those who use it. As in any monarchy, the worker is the subject of the capitalist, having to follow their orders, laws and decisions while on their property. This, obviously, is the total denial of liberty (and dignity, we may note, as it is degrading to have to follow orders). And so private property (capitalism) necessarily excludes participation, influence, and control by those who use, but do not own, the means of life.
[...]
To summarise, anarchists are in favour of the kind of property which "cannot be used to exploit another -- those kinds of personal possessions which we accumulate from childhood and which become part of our lives." We are opposed to the kind of property "which can be used only to exploit people -- land and buildings, instruments of production and distribution, raw materials and manufactured articles, money and capital." [Nicholas Walter, About Anarchism, p. 40] As a rule of thumb, anarchists oppose those forms of property which are owned by a few people but which are used by others. This leads to the former controlling the latter and using them to produce a surplus for them (either directly, as in the case of a employee, or indirectly, in the case of a tenant).
The key is that "possession" is rooted in the concept of "use rights" or "usufruct" while "private property" is rooted in a divorce between the users and ownership. For example, a house that one lives in is a possession, whereas if one rents it to someone else at a profit it becomes property. Similarly, if one uses a saw to make a living as a self-employed carpenter, the saw is a possession; whereas if one employs others at wages to use the saw for one's own profit, it is property. Needless to say, a capitalist workplace, where the workers are ordered about by a boss, is an example of "property" while a co-operative, where the workers manage their own work, is an example of "possession."
Interesting, this libertarian socialism. DA, mind telling me how such a state of affairs could arise in a non-coercive manner?
This is an interesting question, but since it is quite hypothetical it is difficult to answer. So I'll give you some historical examples and you could judge for yourself if they are non-coercive in nature (just a minor note here, Anarchists do not oppose violence against property, for property, as distinct from possession, is inherently illegitimate and thus it is non-coercive to destroy it, and if one feels as though they have been attacked, defending one's self against the attackers is not only allowable under anarchism but encouraged and Capitalistic exploitation could be considered to be such an attack).
The largest anarchist society (in terms of numbers of people) to arise was during the Spanish Revolution, which you can read more about here:
http://en.wikipedia.org/wiki/Anarchism_in_Spain and here
http://en.wikipedia.org/wiki/Anarchist_Catalonia; the longest surviving anarchist society, though smaller in number of participants was the Israeli Kibbutz movement,
http://en.wikipedia.org/wiki/Kibbutz, though it must be noted that it has lost much of what defined it as an Anarchist society today.
Alternatively you could listen to Chomsky describe both in brief here, at about 6:10 :
And for a more recent example of a movement of direct democracy (along with solidarity/mutual aid, this is one of the key forms of praxis that Anarchists advocate for and encourage) that shows signs of anarchist principles being put into action, you may be interested in checking out what has been happening in Venezuela:
http://www.dissidentvoice.org/2008/1...-in-venezuela/
Edit: some of these sources are ones that I posted previously, but considering there relevance and since I didn't get any responses then, I figure I won't get hung out to dry for repeating them.