What are the differences between these two competing ideologies, and more importantly what is the same? Let's answer that question.
Differences: There are not that many differences between Communism and Fascism. In fact the only key difference is their overall approach toward reaching the same goal. Communism is primarily global. It seeks to impart its ideology on a global scale such that everyone becomes equal. Fascism is similar, but instead of everyone in the world being equal economically and socially, Fascism supports nationalism. In essence, Fascism has one group (ideology, religion or ethnicity or race) sitting at the top ruling over all others. In fact Fascism even embraces aspects of Capitalism as true equality is unattainable at both the social and economic levels. As such Fascism embraces some market mechanisms and nationalistic notions that cannot be stopped due to them being part of human nature (i.e. the need to work and do something of value which is why Nazi Germany embraced the ethnic form of Fascism). What is also important to note is that F.A. Hayek in "Road to Serfdom" explained that all Communist countries are doomed to fail as the ideas and concepts of equality fail in a communist system and give way to the nationalist based Fascism ideology. As such there really is only one true difference; one (communism) seeks world total social and economic equality while the other (Fascism) succumbs to the nationalistic notions where there are those greater than everyone else. However, the communist always in the end becomes a Fascist while calling themselves communist in name only.
The same: Almost everything else these two ideologies have in common is exactly the same. They are both based off of the Marxist ideas as portrayed in the Communist Manifesto and Capitilo 1 and 2. Capitalism to both ideologies is the source of all evils. Both look to create some form national government to govern the entire world. Both seek equality amongst the people as much as possible whether it has to be engineered or coerced. As such, all societies that embrace these ideologies have a national education system, a command economy, and either a single ruler or set of rulers at the top deciding what people can and cannot do. The roots of Marxist socialism run deep as people who spout these ideologies typically advocate violence to create their "ideal" societies. Each will have a national retirement system and a method of controlling public information. The list literally goes on forever. Aside from the aforementioned differences, both are almost exactly the same thing.
Conclusion: These are the Communism and Fascism compared. They are in fact parts of the same ideology, the Marxist ideology. There counterparts are Socialism and Progressivism. The ideology of Socialism in its most modern form rejects the idea of ethnic or racial control, but advocates for a benevolent group or individual leading the world over (they look toward the U.N. as that body). It too however still resorts to violence when needed. Progressives are the most advanced model in the modern era and advocate a peaceful transition toward their goal of economic and social equality, but they do this by infiltrating governments and manipulating laws until they get their way. Otherwise these two along with Communism and Fascism are also the exact same as they seek one unified world with social and economic equality. Of course, both in their youth embraced ethnic and racial purity like Fascism, though communism for a time did as well. In the modern era, these labels are really just for show (at least from my point of view). Communist China has fully embraced the same economic system of Nazi Germany, where production is generally allowed to continue so long as the government allows it or approves of it. Nazi Germany was in fact ruled by the National Socialists (this is what Nazi means). So when it comes down to the dirty little details the differences between each are none at all. Some groups want violence to achieve their ends, while others will infiltrate government and transform it slowly into their ideal utopia regardless of who may suffer in the end. Gone is the racial, ethnic and religious purity unless you are talking about small groups like the Neo Nazi's, and other small segmented breakaway groups in each. However, nationalism still comes to the forefront which will always morph the ideology and the government into a Fascist one (though they may be loath to call themselves that). As to the competition between these groups, it has nothing to do with differences in ideology as they both want the same thing. However, the reason why they compete is something more primal, the lust for power (which is why the communist French sided with the Nazi's in WW II, and then later competed against them for control). So in the end you can really just call them what they really are Marxist.