Here is the modern US National Socialist Party's economic stance. As far as I understand, the idea of 'socialism' which led to the continuation of the term past the original Nazis was really a command economy which directed money towards the 'nation,' by which they mean whites.

This may answer your question in ways that a definition leading directly back to the original Nazis may not, because it clarifies a popular conception of socialism which isn't technically correct. This is not a conflation unique to neoNazis, the Mises Institute people also make no distinction beyond command economies and market socialism. I believe Objectivists also, generally, make no allowance for market socialism.

To answer your question in short, socialism is where the workers own the means of production. A democratic government that manages the means of production (command economy) is considered by some to be socialism, so is a market economy where all companies are employee-owned (market socialism). There are other sorts of socialism, but all have the central tenet of eliminating the ownership class, if you will.

Fascism is authoritarian nationalism. It involves the creation of a national narrative which distinguishes a racial in-group, and seeks to create a powerful state to protect and promote the in-group at the expense of the out-groups. As far as I know, all the major fascist movements have been socially conservative, even on issues which are not related to race.

I would definitely say that characterizing NS as a socialism would be a devil's advocate position at best among philosophers, even those which are further right. Particularly, in Nazi Germany, there is no possible way for it to have been a socialism, since businesses were controlled by a nondemocratic state. However, this is using a proper definition of socialism which differs from the one many people use.

Another way of saying that would be, there's probably no socialist alive who thinks having a dictator pulling the strings of the economy is a good idea, no matter who that dictator is.

One major distinction has not yet been mentioned:

Socialism asserts that one's class is what unites people into a social whole, while fascism asserts the primacy of blood. Socialism is this, at least in theory, very internationalist. There's no categorical difference between a Chinese or a Dutch (factory) worker. There is however a categorical difference between a Dutch worker and a Dutch factory owner ("capitalist").

Fascism does not acknowledge this distinction. What matters is the Volk, the people, the body politic of the nation, et cetera, et cetera. The Dutch factory worker and the Dutch capitalist are both different parts of the same organic whole. Categorically distinct from all non-Dutch.

In theory, this means that facists are in favour of the logical conclusion of the nation-state, while for socialists the nation-state is just another instrument in maintaining class structures that oppress the proletariat; and should thus be overthrown. Hence the slogan: "Workers of the world, unite!"

(Although, admittedly, the Nazi's allowed for different "categories" of Germanicness, but that relates more to their weird philosophical beliefs in an Aryan race and their various "Nordic" successor peoples.)

This all depend on who's form of fascism you are referring to and who's form of socialism you are referring to. For the sake of this answer, I will take Italian-Greek fascism and Marxist German socialism.

In regards to economy, they are of course different. Socialists believe in taxing the rich, government intervention in trade and having the farmers own their own goods and supply.

Fascists however see the rich as an asset. The fascists, unlike the socialists, believe in seizing ALL factories and farms for the government. I understand that some socialists/communists believe in doing this as well, but I am referring to Marxism, not Stalinism.

In terms of social equality, the socialist ideology leads. Under most forms of pure socialism, every race, religion, sex and creed is equal. Unlike socialism, fascism usually picks one race and religion to rule as the ideal person to represent their nation in both idea and politics. This can and has lead to the deportation and or execution of "unwantables".

National Socialism is the odd mixing of these two ideologies. It is where two ideologies that are opposites in most retrospect collide into one authoritarian rule. It's similar to how Socialism and Anarchy have sometimes collided in some forms of Marxist Communism. Very odd.

Now for National Socialism as a comparison of the other two ideologies.

National Socialists, like the fascists, seized all factories, goods, ect. However, unlike the fascists, national socialists often hate the rich and will take them from power. This was seen in Nazi Germany ... however, this was only in an ideological form. The people thought that they were getting rid of the greedy aristocrats, when Adolf Hitler was really only targeting Gypsies, Jews and Slavs, blaming them for all of their countries hardships. Like fascism, National Socialism also picks one race and religion to lead the nation. In Nazi Germany's case, it was of course German Christianity.