In practical terms, though, the corporations were basically nationalized, with the former owners being made middle managers (Betriebsführers) at best. Nearly all important decisions were made by the government, especially relating to wages and prices. There weren't labour unions, but the economy was sufficiently controlled that any shopowner that charged too much or paid workers too little would be arrested pretty quickly, though this was basically limited to small businesses that the government didn't control outright. In practical terms, Germany was nearly as left-wing in economic policy as the Soviet Union was.
That is right. The really important difference between Soviet-style communism and National Socialism (and similarly Italian Fascism) nobody mentioned yet is that the Nazis didn't touch
private property, with two notable exceptions: a) some industries of strategic importance for the war effort and b) whatever they stole from the Jews, political opponents, other victims and in the occupied territories during the war. In practice, while you were subject to heavy regulation, you would keep ownership and control of any businesses or real estate you owned, while the Soviets would let you keep nothing. While your economic freedom was somewhat limited in NS Germany, you could buy or sell property or start and operate a business. This distinction is important because it is why a) the Soviets thought of Nazism/Fascism as just another manifestation of Capitalism and b) why the West initailly tried to appease the Nazis as potential allies against Communism (because while they weren't democrats, they at least weren't socialists).
They were socialist (although communist is a closer word; there is no way they would use it in their name) in the sense they wanted a single state government to be in control, stripping the private sector of the means of production and commerce, and the opposition of any power; the consolidation of power to the state.
Where they are different is what they do with that power. Whereas a communist government would work (or at least should work) for the equality of all men, a Nazi one worked against it. It established and strengthened inequality, made it pervasive, worked to enforce it. The Nazis and those of a similar mind worked to establish and maintain their hierarchies: one race above another, one caste above another, one profession above another, and above all the state and it's rulers.
There are some misconceptions in this. As I said above, the Nazis left the private sector be, as long and as far as it was in their interest to do so, while in communism there is no private sector because there is no private property.
The hierarchies the Nazis established were ones of loyality to their cause, they didn't really respect the existing hierarchy however. That's why they and the formerly powerful aristocracy often hated each other. To the prussian aristocrats and the rich industrialists that had dominated german society, the Nazis were initially just another proletarian mob, though many changed sides later on. For the average german citizen social mobility did still exist like before Nazi rule, even if political views could help or hurt tremendously, there was in no way a caste system or something like that, with the obvious exception of the racial aspect.