What's the difference between 'fascism' and 'socialism'?

The question “What’s the difference between fascism and socialism?” should really be “What are the differences between fascism and socialism?” After all, there are many—not least because throughout history, there have been many different kinds of fascist and socialist movements, philosophies, and governments. Nevertheless, fascism and socialism have enough characteristic aspects, tendencies, and beliefs (usually diametrically opposed) to make their casual or interchangeable use confusing, if not misleading.

Fascism

While fascism is often used broadly to refer to any tendency toward (or exercise of) autocratic or dictatorial control, historically it refers to a political philosophy, movement, or regime with some key traits. In general, fascism is characterized by extreme nationalism; enforces strict social, economic, and often racial hierarchies; is corporatist and imperialist; supports autocratic control of the government (as by a dictator); promotes strict traditional gender roles and military values; and forcibly suppresses opposition. Fascist movements throughout history have also been typified by their opposition to Marxism, democracy, and political and cultural liberalism.

Fascism today might not look exactly as it did in the 1930s, but refugees are once again on the road everywhere. In multiple countries, their plight reinforces fascist propaganda that the nation is under siege, that aliens are a threat and danger both within and outside their borders. The suffering of strangers can solidify the structure of fascism. But it can also trigger empathy once another lens is clicked into place.
— Jason Stanley, How Fascism Works: The Politics of Us and Them, 2018

Socialism

Since the term socialism entered English around 1830, it has consistently referred to a system of egalitarian social organization in which private property (not to be confused with personal property) and the distribution of income are subject to social control. The conception of that control, however, has varied, and socialism has been interpreted in widely diverging ways, ranging from statist to libertarian, from Marxist to liberal. In the modern era, “pure” socialism has been seen only rarely and usually briefly in a few Communist regimes. Far more common are social democracies, such as Sweden and Denmark: democratically elected governments that employ some socialist practices but within a capitalist framework in the belief that extensive state regulation paired with limited state ownership produces a fair distribution of income without impairing economic growth.

In my view, the most fundamental characteristic of socialism is its commitment to the creation of an egalitarian society. Socialists may not have agreed about the extent to which inequality can be eradicated or the means by which change can be effected, but no socialist would defend the current inequalities of wealth and power. … To varying extents, all socialists have therefore challenged the property relationships that are fundamental to capitalism, and have aspired to establish a society in which everyone has the possibility to seek fulfilment without facing barriers based on structural inequalities.
— Michael Newman, Socialism: A Very Short Introduction, 2005