The short answer is that no one really knows. Historians have posited several theories. Many historians point to Daly's support for William A. Clark's opponent in the 1888 campaign for territorial delegate as the beginning of the feud. They speculate that Daly supported Republican Thomas Carter because he believed a "Republican would have greater leverage with what Daly ... presumed to be a Republican administration in 1889, and that his leverage could and would be used to squash or at least slow the prosecution of the indictments filed by Democratic President Grover Cleveland's administration against Daly and his associates' illegal timber cutting operations." (Montana Heritage, 153-54.) This is the explanation presented in the textbook.
Historian David Emmons, on the other hand, suggests a cultural explanation. In "The Orange and The Green in Montana: A Reconsideration of the Clark-Daly Feud," Emmons points out that Clark was of Scotch-Irish descent (an "Orange") while Daly was an Irish Catholic (a "Green"). According to Emmons, Daly was very active in Irish Catholic nationalist organizations and primarily hired Irish Catholic immigrants. Clark, by contrast, primarily hired Protestant Cornishmen from England. According to this theory, the "bitter enmity between Northern Ireland Protestants and Irish Catholics" defined the two men's feelings toward one another (Montana Heritage, 154).
Most likely, cultural, economic, and political factors all contributed to the feud, with personality probably playing a role as well.
Among the best books on the Clark-Daly feud is Battle for Butte: Mining and Politics on the Northern Frontier, 1864-1906, by Michael P. Malone (Seattle: University of Washington Press, 1981). Emmons's article, "The Orange and the Green" was originally published in Arizona and the West, 28 (Autumn 1986): 225-45 and reprinted in Montana Heritage: An Anthology of Historical Essays, edited by Robert R. Swartout Jr. and Harry W. Fritz (Helena: Montana Historical Society Press 1992).