One of these is a motive. In the poem, Salieri has to inform Mozart of his own talent. Hardly the "conceited brat" of the movie, Mozart innocently runs some of his music by his older rival, who, unable to contain himself, exclaims "what depth, what profundity!" Mozart seems not to get the point. "What, is it good?" he asks. "My God, Mozart" returns Salieri. "You don't deserve yourself!" More to the point, Mozart doesn't even recognize his own genius. Salieri alone knows it ("It's I who know!").
Given that Salieri was the more successful composer, lauded by all Vienna, favored by the emperor, and with a string of box office bonanzas to his credit not just in Vienna, but in Paris and Italy as well, one might wonder what could have motivated a man like that to want to kill a less successful rival. There was, after all, enough room on the operatic stage for both of their productions (one composer couldn't write more than an opera or two in a single year, and the needs of the public were enough to keep several in business). What Pushkin does is to answer that question, and to do it well enough that what might seem like a long-shot answer is no longer even questioned by the public today. Of course he did it because he alone knew that he wasn't the greatest composer and it was killing him! The rest of Vienna might have thought he, Salieri, was the greater, but he knew the real truth, and he couldn't stand it!
I'm reminded of an essay by Mark Twain ("What is Man?") in which he makes plain that it is the desire to secure one's own approval that takes precedence over everything else; that even when rescuing another person from drowning, it is not an act of altruism stemming from the desire to keep another from harm, but because one's own scheme of values, one's own inner conscience, requires it. In the end, says Twain, we are answerable first and last to our own inner psychological needs. Thus, it didn't matter what Vienna thought, it mattered what Salieri thought.
Since the drama that unfolds is all in Salieri's mind, there isn't any need for historical evidence to back it up. And it has the genius of transferring the drama from out in the material world of things and events and into the theater of Salieri's own conscience, and, by extension, into ours.
Because all of us, as the movie will make clear, are Salieri.