What is the main conflict in Amadeus?

On the face of it, the main conflict of Amadeus is between the musical abilities of the composers Salieri and Mozart. Salieri has devoted his life to music and risen to the rank of court composer. He is highly respected and has established a solid reputation in his field. Mozart is a former child-prodigy who blazes into the music scene in Vienna and rapidly becomes the emperor's favorite composer, destabilizing the career Salieri has worked so hard to build.
When Salieri introduces Mozart, he recounts Mozart's childhood as a "wunderkind" who toured the courts of Europe, astonishing everyone with his talents. Mozart's father, Leopold, devoted himself to nurturing his son's genius. Salieri's own father, by contrast, had no interest in Salieri's talent, and it wasn't until he died that Salieri was able to pursue studies and later a career in music. Mozart was thus provided from birth with opportunities to develop his talent that Salieri had to wait and work hard to achieve.
This knowledge makes it easier for Salieri to accept Mozart's talent and seems at first to give him the impression that, had he had the same opportunities as Mozart, they would be musical equals. When he finally meets Mozart, he is chagrined to discover that Mozart is a genuine genius, whose facility with music is so far beyond Salieri's that Salieri cannot hope to ever match him. This is evident in many scenes but perhaps most strikingly when Salieri, astonished, asks Mozart's wife whether the musical scores she has procured for him are "originals," to which she replies "Yes, sir; he doesn't make copies." Salieri is staggered to realize that Mozart does not even need to draft his music when composing it, and it is at this point that he begins to understand the gulf that separates their respective abilities.
However, the real conflict is less about the men's music and more about their personalities. Mozart is many things Salieri is not: he is popular, especially with women; he is modern where Salieri is safely traditional; and he is highly charismatic, which enables him to break rules repeatedly and get away with it. He swears in front of the emperor, includes a ballet in his opera when ballet is illegal, and baldly states his opinions without respect to the senior members of the court. He is rude to his social betters and keeps company with "commoners." He also has a reputation as a drinker, a womanizer, and a spendthrift. Despite these many breaches of etiquette, Mozart remains the toast of Viennese society, by whose rules and customs Salieri strictly abides.
Even Mozart's abrasive personality might have been acceptable to Salieri if he had shared Salieri's passion for music, but he does not. Despite his talent, or perhaps because of it, Mozart seems to Salieri to be as careless about music as he is about everything else. Salieri has given his life to music and compared to Mozart, he knows he is only a mediocre composer. Mozart, by contrast, seems to waste most of his time in "vulgar" pursuits and yet somehow still produces works of genius. It torments Salieri that Mozart is so gifted and so apparently ungrateful for his gifts, and this is the reason Salieri decides to torment Mozart in turn—to punish him for his lack of appreciation.
Ultimately, it transpires that Mozart has never felt himself to be in conflict with Salieri and has, in fact, always considered Salieri a friend. It therefore can be argued that the main conflict in Amadeus is not between Salieri and Mozart but between Salieri and his personal demons of jealousy, ambition, and vindictiveness, which he only overcomes as Mozart lays dying.

The main external conflict in Amadeus is between Mozart and Salieri. Salieri has been court composer for quite some time. He is very talented, albeit somewhat short of inspiration. However, his work is generally admired, if not exactly loved. Then along comes Mozart, this brash, arrogant, supremely gifted genius who seems to have fallen straight from the heavens. Soon, the young pretender is the talk of the town. He is feted wherever he goes and celebrated as the "Next Big Thing." All of a sudden, no one is interested in Salieri's music anymore. His work seems so dull, so pedestrian, so frightfully passé, at least in comparison with the sublime compositions of Amadeus.
At the same time, there is also a very important conflict in the play between art and taste. Just as quickly as Mozart rises, so does he fall. His star rapidly fades, and his work is increasingly ignored by the Emperor and his court lackeys. As they set the tone for what is fashionable in society, Mozart soon becomes almost as out of date as Salieri. Salieri, though rather pleased to see this wunderkind come crashing down to earth with an almighty bump, does still recognize his young rival's genius. Though they never truly become bosom buddies, the two composers do develop a certain solidarity with each other in that they both know the difference between great art and mere hackwork. The problem for Salieri is—no matter how hard he tries—he simply cannot elevate his own work above the level of the second-rate. That is one conflict he can never resolve.