Socialism is a complete unification of state and economy. That's what "social ownership of the means of production" means. The government controls the economy in most scenarios (though libertarian socialism is a thing).

Facism is similar, but it usually has a market economy to some extent with the government only seizing what it needs to accomplish its goals. It'll likely eventually seize pretty much everything, but that's not the end goal. For example, in Nazi Germany, the government seized the car dealerships to make tanks and whatnot, but probably didn't seize the local bakeries and whatnot because there was no need to (unless it was run by an enemy of the state, like Jews or whatever the flavor of the week is).

The goal with facism is to pursue the goals of the state, which is usually to subjugate enemies. The goal for Nazi Germany was to become the most powerful country in the world, and to get the necessary power, they blamed all of their problems on "undesirables" (in this case Jews, homosexuals, and other minorities), arguing that if they eliminate those enemies, Germany would be greater. Once those enemies are exhausted, they'd find a new minority to blame their problems on.

The goal of socialism is equality for its people. To enforce that, it needs control over the economy.

They use similar tools to meet their goals, but that doesn't mean either is defined by a union of business and the state, it just means that's a common tactic they use.

Facism is defined by hate against some minority in order to get power to unify some definition of "the state." That's it. If it can do it without controlling the economy, fine, it's just another tool in its toolbox.