Fascism specifically centers around a hierarchy based on race, ethnicity, place of origin, sex, gender, sexuality, or some other (for the most part immutable) characteristic of a person. It also may specify heavy corporatism (as what governments like Mussolini's Italy structured their economy around).

Leftism centers around abolishing all unjust hierarchies, including those that fascism relies upon. It is also anti-capitalist and, obviously, anti-corporate.

"Fascism" doesn't just mean "authoritarianism". Fascism doesn't even strictly need a state – it's mostly social and economic in its nature, and doesn't say anything about the structure (or existence) of government. Anarcho-capitalism, for instance, starts to decay into fascism, where there may be no "government", but rather private entities (like corporations and individuals) restricting or blocking the social and economic participation of certain groups based on a social gradient, or in general depriving people in those groups of rights (like enslaving, harming, or killing them, denying them food or healthcare, etc.). There are always enforcers of fascism, as it's an inherently unequal and oppressive ideology, but whether the oppressors' power ultimately comes from governmental organizations or non-governmental organizations doesn't matter. You could argue this does constitute authoritarianism, and I wouldn't disagree, but my point is that "big government" and fascism are entirely different concepts.

For the most part, fascism can be considered an ideology of emphasizing a supposed "former glory" of a nation or peoples, which co-opts socialist critiques of capitalism and twists them to emphasize immutable characteristics like ethnicity or masculinity as being the cause of economic woes, rather than class; Fascism, while taking significant influence from leftist ideology in its rhetoric, turns it on its head and repurposes it for the "Volk" (some population/identity based on generally immutable characteristics) rather than the worker. It ends in the dismantling of trade unions and other leftist structures, and an economy comprised of corporatoid organizations which is kept afloat by the constant drive for "purification" (the enforcement of a bigoted hierarchy) which never concludes, resulting in the gradual narrowing of who is included in the "in-group" (cannibalizing itself) after a certain point.

Leftism puts class warfare above all else, and while some leftists could incorporate fascist elements into their beliefs – that being, social conservativism, as elements like misogyny and homophobia aren't impossible to find in the belief systems of self-identified communists (mostly apologia for the errors of authoritarian communist governments) – the socioeconomic structure of socialism compared to fascism is so radically different that it's impossible to fit full-on fascism into a socialist structure. Fascism praxis perhaps may be observed as "welfare for a very specific class of peoples, reliant on the oppression of lower classes of peoples", where the "out-groups" are forced into to the lowest classes, and the "in-groups" who are of lower classes may see a limited amount of welfare. Fascism combines class-based hierarchies with "they're different than me"-based hierarchies; this very stark class division and exploitation of lower classes completely conflicts with core leftist ideology.