Ok, there is a large literature on this, and it really depends on who you ask. Part of the problem is that defining fascism is very difficult. Invariably, scholars take two approaches: either too broad or too narrow. The problem comes with how one differentiates between fascism and a more normative authoritarian dictatorship. A typical example is how to treat Franco in Spain. There was a fascist party there, the Falange, but they were not very powerful. Another issue is that many scholars and lay-people view Hitler's Germany as the archetypal fascist state, even more so than Mussolini's Italy. This is, of course, problematic because Mussolini was the one who founded the first "fascist party" and is the one who came up with most of its tenets. Anyway, directly onto the question asked.

I'll start with the most basic and let others fill in the gaps.

Fascism comes from the name of a movement founded by Benito Mussolini in Italy in the late 1910s and early 1920s. It was formally incorporated into the National Fascist Party in 1921, and that party, under Mussolini's leadership, would gain power in Italy in 1922. Mussolini was only deposed in 1943, during World War II.

The only real connection Fascism has to Ancient Rome is in the imagination of the fascists themselves and the fact that both groups were based in Italy. While it's true that the "fascii" were used in Rome as symbols of power, and that Mussolini was "looking back" to Rome in using this name for his party, that does not mean that there was any real relation between the two beyond the ephemeral. The Romans were not fascists in any sense, nor did the Fascists proclaim to rule in the Roman fashion (i.e. using Roman customs or law). Rather, they proclaimed that Fascism would bring back the glory of Ancient Rome to modern Italy. And that's the extent of the connection.

A few sources that deal with fascism specifically:

Ben-Ghiat, Ruth. Fascist Modernities: Italy, 1922-1945. Berkeley: University of California Press, 2001.

Eksteins, Modris. Rites of Spring: The Great War and the Birth of the Modern Age. Boston: Houghton Mifflin, 1989.

Falasca-Zamponi, Simonetta. Fascist Spectacle: The Aesthetics of Power in Mussolini’s Italy. Berkeley: University of California Press, 1997.

Ferguson, Niall. The War of the World: History’s Age of Hatred. London; New York: Allen Lane, 2006.

Legge, Jerome S. Jews, Turks, and Other Strangers: The Roots of Prejudice in Modern Germany. Madison, Wisc: University of Wisconsin Press, 2003.

Mangan, J A. Superman Supreme: Fascist Body as Political Icon - Global Fascism. London: Frank Cass, 2000.

Mazower, Mark. Dark Continent : Europe’s Twentieth Century. 1st American ed.. New York: Knopf, 1999.

Mosse, George L. The Image of Man: The Creation of Modern Masculinity. New York: Oxford University Press, 1996.

Orlow, Dietrich. The Lure of Fascism in Western Europe : German Nazis, Dutch and French Fascists, 1933-1939. New York: Palgrave Macmillan, 2009.

Paxton, Robert O. The Anatomy of Fascism. 1st ed. New York: Knopf, 2004.

Payne, Stanley G. A History of Fascism, 1914-1945. Madison: University of Wisconsin Press, 1995. Thayer, John Alden. Italy and the Great War; Politics and Culture, 1870-1915. Madison: University of Wisconsin Press, 1964.

A few sources on the German side that discuss Nazism (which I'm much more familiar with):

Allen, William Sheridan. The Nazi Seizure of Power: The Experience of a Single German Town, 1922-1945. Rev. ed.. New York: FWatts, 1984.

Ames, Eric, Marcia Klotz, and Lora Wildenthal. Germany’s Colonial Pasts. Lincoln: University of Nebraska Press, 2005.

Bessel, Richard. Nazism and War. New York: Modern Library, 2004.

Betts, Paul, and Greg Eghigian. Pain and Prosperity : Reconsidering Twentieth-Century German History. Stanford, Calif: Stanford University Press, 2003.

Conrad, Sebastian. German Colonialism: A Short History. Cambridge; New York: Cambridge University Press, 2012.

Eley, Geoff. Reshaping the German Right: Radical Nationalism and Political Change after Bismarck. New Haven: Yale University Press, 1980.

Frevert, Ute. A Nation in Barracks : Modern Germany, Military Conscription, and Civil Society. 1st ed.. Oxford ; New York: Berg, 2004.

Fritzsche, Peter. Germans into Nazis. Cambridge, Mass: Harvard University Press, 1998.

———. Life and Death in the Third Reich. Cambridge, Mass: Belknap Press of Harvard University Press, 2008.

Koonz, Claudia. The Nazi Conscience. Cambridge, Mass: Belknap Press, 2003.

Kühne, Thomas. Belonging and Genocide: Hitler’s Community, 1918-1945. New Haven: Yale University Press, 2010.

Lerner, Paul Frederick. Hysterical Men : War, Psychiatry, and the Politics of Trauma in Germany, 1890-1930. Ithaca, NY: Cornell University Press, 2003.

Liulevicius, Vejas G. The German Myth of the East: 1800 to the Present. Oxford; New York: Oxford University Press, 2009.

———. War Land on the Eastern Front: Culture, National Identity and German Occupation in World War I. New York: Cambridge University Press, 2000.

Mazower, Mark. Dark Continent : Europe’s Twentieth Century. 1st American ed.. New York: Knopf, 1999.

———. Hitler’s Empire: How the Nazis Ruled Europe. New York: Penguin Press, 2008.

Peukert, Detlev. Inside Nazi Germany: Conformity, Opposition and Racism in Everyday Life. London: Batsford, 1987.

Raeff, Marc. “The Well-Ordered Police State and the Development of Modernity in Seventeenth- and Eighteenth-Century Europe: An Attempt at a Comparative Approach.” The American Historical Review 80, no. 5 (December 1975): 1221.

Rahden, Till van. Jews and Other Germans: Civil Society, Religious Diversity, and Urban Politics in Breslau, 1860-1925. Madison, Wis: University of Wisconsin Press, 2008.

Smith, Helmut Walser. The Continuities of German History : Nation, Religion, and Race across the Long Nineteenth Century. Cambridge ; New York: Cambridge University Press, 2008.

Weitz, Eric D. Weimar Germany : Promise and Tragedy. Princeton, NJ: Princeton University Press, 2007.

EDIT: sources formatting