Animosity between Czechs and Slovaks on the make-up of the Czechoslovak state were an issue for a good deal of the twentieth century. Just for the context, we'll start in the late 19th century, when the Czech and Slovak lands were still a part of Austria-Hungary. Although both Czechs and Slovaks were Slavic peoples with a high degree of mutual ineligibility between the Czech and Slovak languages, the Czech lands had been in the administrative jurisdiction of Austria, while Slovakia was in the Hungarian part of the empire. Knowing this is important to understand the ideology of Slovak nationalists, who identified histrionically more with Hungary and believed in a strong Catholic tradition (see Jehlička), especially compared to the Czech's Protestant national identity, which revolved around icons such as Jan Hus, Jan Žižka, and the Battle of the White Mountain (Bílá Hora).

During the Paris Peace conference in 1919, Tomáš Masaryk, Edvard Beneš, and Milan Štefánik, the Czech and Slovak peoples' main representatives at the conference, understated the differences between Czechs and Slovaks and advocated for the creation of a Czechoslovak state. This was primarily due to the belief that independent, small nations in Central Europe would have trouble existing between the cultural and political giants of Germany and the Soviet Union. This idea stemmed partly from Austro-Slavism, an ideology popular among Czech intellectuals during the late 19th and early 20th centuries that supported cooperation among the Slavs of Austria-Hungary in order to gain political and cultural "ascendancy" within the dual Monarchy. Beneš especially had been an adherent to Austro-Slavism, and "All Slav cooperation." As alluded to previously, prominent Slovak nationalists, such as Andrej Hlinka and Francis Jehlička, were diametrically opposed to a Czechoslovak state and desired instead for an independent Slovakia. Fast forward to late 1938 during the Munich crisis, where Germany annexed the Sudetenland from Czechoslovakia (Czechoslovakia's borderlands with Germany and Austria that were populated primarily by German speakers), and Slovak politicians took the pressure on the Czechoslovak government to lobby successfully for increased federalism and Slovak rights, which became reflected in the nation's new name, the Second Czecho-Slovak Republic (note the hyphenated "Czecho-Slovak" is grammatically incorrect in both Czech and Slovak languages). In 1939 after Germany occupied the rest of the Czech lands, Slovakia declared itself independent as an axis aligned state called the First Slovak Republic, led by Jozef Tiso. After the reunification of Czechoslovakia after the war, there was still a bit of national resentment, and President Beneš had briefly considered punishing Slovaks in a similar way as Germans and Magyars, but ultimately decided against it (see Gronský, pages 54 and 55 for German and Magyar post-war punishments in the "Beneš Decrees," mainly regarding the confiscation of privately owned agricultural and industrial property).

During the Socialist period, Czechs had been much more publicly outspoken in protests against the Communist Party and Warsaw Pact dynamic than Slovaks (per Žantovský, out of 241 original signators of the human rights document Charter 77, only six were Slovak). Furthermore, after 1989, there were a number of incidents that helped to grow Czech and Slovak resentment towards each other, although mostly Slovak resentment directed at the Czechs. The new President, Václav Havel (a Czech), in an attempt to rebrand the nation's image abroad announced that Czechoslovakia would cease to supply "arms to every bloody dictator and every terrorist group around the world." Prior to this, Czechoslovakia had been the fourth largest arms manufacturer and exporter in the world, and most of the arms industry was located in Slovakia (although Bohemia is historically the industrial heartland of Czechoslovakia, Warsaw Pact politicians wanted Czechoslovakia's arms manufacturing further from their border with the West). The arms industry had brought a good deal of jobs and overall prosperity to Slovakia, and so Havel's choice appeared to be an attack on the Slovak people directly, despite the fact that with the end of the Cold War, the market for arms had decreased anyway, and Havel became widely unpopular in Slovakia. Other issues between Czechs and Slovaks included how to orient foreign policy after the Cold War, whether or not to change the state's name in order to better reflect the two distinct ethnic groups that made up the state, and what degree of federalism should be implemented in politics. At the time, the Slovak Christian-Democratic party argued that independence in 1918 for Czechoslovakia had been illegitimate due to violations of pledges to the Slovaks that were made at the Paris Peace conferences, and the Czechs would have to completely reavow and would have to reorganize the state to have Czechs and Slovaks on and equal and equitable basis in Czechoslovak law.

In the elections of June 1992, the Movement for Democratic Slovakia party, a party led by Vladimír Mečiar and which supported Slovak autonomy, made huge gains. At the same time, there was a grid lock in parliament on how to re-write the constitution to better represent differing Czech and Slovak interests. Additionally, at this point, Czech and Slovak politicians were also acutely aware of the contemporary wars in former Yugoslavia that had resulted from failed referendums for ethnic autonomy, and wanted to avoid a conflict of the same kind in Czechoslovakia at all costs. So with the backdrop of growing support for Slovak independence, constitutional grid lock, and the desire to avoid an ethnic conflict, Havel decided he would not attempt to intervene in Slovak independence, thinking: "rather than to live for years in a non-functioning federation or in some kind of a pseudo-federation, which is only a burden and a source of complications, it is better to live in two independent countries." The Slovak national parliament passed a declaration of sovereignty on July 17th, which put the validity of the Slovak Constitution above that of Czechoslovakia's federation, effectively declaring independence.

Just a fair warning, I'm writing this from a decidedly Czech point of view. Also, this is my first post in the sub, so mods, I'll be happy to share my credentials if necessary! I have a few more citations I could add that are on a flash drive of mine, but I'd have to look for that later.

edit: forgot to specify "the Cold War" at one point

Works of interest:

Beneš, Edvard. Memoirs of Dr. Eduard Beneš: From Munich to New War and New Victory. Translated by Godfrey Lias. Cambridge: The Riverside Press, 1954.

Gronský, Ján. Dokumenty k ústavnímu vývoji československa II., 1945-1968. Prague: Karolinum Press, 1999.

Jehlička, Francis. Father Hlinka’s Struggle For Slovak Freedom: Reminiscences of Professor Francis Jehlička, Chairman of the Slovak Council. London: The Slovak Council, 1938.

Kuklík, Jan. Czech Law in Historical Contexts. Prague: Karolinum Press, 2015.

Mastný, Vojtěch. “The Beneš-Stalin-Molotov Conversations in December 1943: New Documents.” Jahrbücher für Geschichte Osteuropas Volume 20, Number 3 (September 1972), 367-402.

Taborsky, Edward. President Edvard Beneš: Between East and West, 1938-1948. Stanford: Hoover Institution Press, 1981.

Vyšný, Paul. Neo-Slavism and the Czechs, 1898-1914. Cambridge: Cambridge University Press, 1977.

Žantovský, Michael. Havel: A Life. New York: Grove Press, 2014.

Žantovský, Michael. “In Search of Allies: Vaclav Havel and the Expansion of NATO.” World Affairs Volume 177 Number 4 (November/December 2014), 47-58.