Marx advocated the use of a state to transition society to a stateless condition. This period of socialism, the dictatorship of the proletariat, is the authoritarian streak that led to the Marx/Bakunin break.
Lenin, and probably others, advocated for the use of a state to transition, not Marx. The dictatorship of the proletariate is a transitional period when the proletariates take over the means of production through revolution. There are still classes at this point, hence transition, but the ruling class is the working class. In other words, a "dictatorship" of the many and not the few.
As far as the split goes, it was a lot more involved than just one issue. The problem you are referring to, however, had more to do with using politics of the time in order to motivate workers which they both understood in many parts of Europe were not ready for what they wanted. Marxist were ok with participating in politics, Bakunin was not. They both agreed though that real change would come from an uprising of people and not through political action.
Again, your understanding of Marxist political theory is grossly inaccurate. This is [still] the mythical version of Marxism that was parroted by Soviets, [at least some] Chinese, and Western powers for their own ends--be it masquerading as "true" Marxists, or scaring the shit out of people into seeing Marxism/socialism/communism, instead of its authoritarian practitioners, as "evil". You also appear to misunderstand entirely Marx's conception of the dictatorship of the proletariat. It was in no way authoritarian in any of Marx's explications of the idea.
Marx was--in his political theory, as well as his political activity--focused on defending and advancing the idea of democratic authority of the general interests of the masses. He did not, unlike Bakunin and others, view the masses as mere cannon fodder--Bakunin actually called them this--to throw at standing power to succeed in revolutionary overthrow. Moreover, Marx ardently and unfailingly argued and advocated that the ultimate end, with regard to the masses, was to teach them to walk by themselves, eschewing the bureaucratic regulation through which they are taught from childhood onward to believe in and acquiesce to the authority of those set over them. Workers are to engage the process of self-emancipation through taking charge of and reorganizing society--through bottom-up democratic means. Bakunin and others, on the other hand, argued that teaching the people anything was stupid, and they should merely be convinced to revolt. To this, Marx once wrote that inciting workers without offering them any guiding ideas or constructive self-emancipatory doctrine was "equivalent to vain dishonest play at preaching which assumes an inspired prophet on the one side and on the other only the gaping asses."[1]
Writing during his time of participating in the International, Marx believed it “the business of the International Working Men’s Association to combine and generalize the spontaneous movements of the working classes, but not to dictate or impose any doctrinary system whatever.”[2] He vehemently opposed the moves by Bakunin and others to create secretive, autonomous groups within the International, because such a group “is opposed to the development of the proletarian movement because, instead of instructing the workers, these societies subject them to authoritarian, mystical laws which cramp their independence and distort their powers of reason.”[3] Marx explained to Wilhelm Blos that when he and Engels joined the Communist League, they “did so only on condition that anything conducive to a superstitious belief in authority be eliminated from the Rules.”[4]
The dictatorship of the proletariat is a bottom-up democratic seizure of power. Marx did not have any problem playing politics. He just wanted to see the workers in control of politics, instead of it residing in the hands of capitalists and their elitist servants who would use political power to perpetuate the alienation and division of workers, thus preserving the privileged status of those with capital within the social and political system. Of course, this is also heavily integrated with taking control of the means of production and placing it into the hands of the working masses, as tarentel mentions in a sibling comment. Nevertheless, you are imbuing Marx's notion of "dictatorship of the proletariat" with post-Marx, 20th-century infused notions of dictatorship--that is, authoritarian rule of one (or few) over all.
Feel free to say if you are still convinced of the rightness of your inaccurate understanding and I will offer more explanation.
[EDIT]: For full disclosure, the particular arc and inclusion of quoted items 2-4 is borrowed from David Adam's "Marx, Bakunin, and the question of authoritarianism"[5]. It got directly to the bits I wanted to cover, and saved me the time of having to comb back through my own library to find the relevant quotes I was looking for.
[1]: Francis Wheen's Karl Marx: A Life, 104.
[2]: Marx's "Instructions for Delegates to the Geneva Congress" in Political Writings Vol 3, 90.
[3]: Marx's "Speech on Secret Societies" from Collected Works Vol 22, 621.
As far as the split goes, it was a lot more involved than just one issue. The problem you are referring to, however, had more to do with using politics of the time in order to motivate workers which they both understood in many parts of Europe were not ready for what they wanted. Marxist were ok with participating in politics, Bakunin was not. They both agreed though that real change would come from an uprising of people and not through political action.