These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Awesome Music: Mozart was arguably the greatest composer in history, and Salieri was no slouch himself. Both of their music—but especially Mozart's—is featured throughout the film.
Subverted in one case: Salieri's opera was fantastic, and was played completely straight, but numerous characters are shown yawning through it.
One could argue that it was perhaps the way Salieri remembers that opera. Or that he was focusing on the worst among the audience. It wouldn't be surprising, considering he thought himself to be so inferior to his rival.
As an explanation of the opera, what Salieri is doing here is basically catering to the lowest common denominator. He's giving the audience a big bang at the end of the show so that they know when to applaud. The sound mix deliberately emphasizes this; all the instruments mix together into an overwhelming blur of sound with no clarity to it. It's not bad, just unimaginative. Salieri hates himself for not being able to do better than this.
At the beginning, Salieri plays snippets of his music to the confessor, who admits never hearing them before. When he plays a third snippet (the beginning of "Eine Kleine Nachtmusik"), the priest exclaims "Oh I've heard that" only for Salieri to reveal that third bit was Mozart's...
Family-Unfriendly Aesop: Talent and character are completely unrelated. A handful of rare individuals are born special, for no reason or merit, a few others are sharp enough to see talent in others but do not have it themselves and are doomed to carry the burden of mediocrity. And most people are just mindless morons.
It could be that some people are more obscenely talented than others, but you still shouldn't let envy get the better of you and let it take over your life.
Perhaps the best message to take from it is "Bitterness and envy will destroy You and comparing Yourself to others is a waste of Your own talents". Though don't piss off God is also a good message.
Yet some more readings are possible: that blind obeisance and faith can cause disappointment and even cause people to do the wrong thing, or at the very least Salieri's expectations about God were misplaced - i.e. he expected to be "given" his talents in exchange for his good services, while the reality is that we have no way of knowing who can be Mozart.
Why is the title of the movie "Amadeus" when it is actually about Salieri? Perhaps it is because even in the name of the film about him, Salieri is pushed aside in favor of the more famous and successful Mozart, whose freaking middle name is so recognizable that anyone who hears it will immediately think of him.
Amadeus also means "God's beloved". Salieri's Blasphemous Boast makes more sense now, doesn't it? (This point also explains the odd fact that the play is named after Mozart's middle name and isn't called Wolfgang).
How did Salieri know the parts of his story that he wasn't there for and had no ear to listen for him?
He's had several years since Mozart's death, he probably has picked up enough gossip over the years to figure most of it out.
Notice how some of the operas are translated into English (The Magic Flute and its parody) while the others are left in the original Italian? It's because those others were written in German, which is of course the language of conversation in Vienna. A very novel use of the Translation Convention.
Fridge Logic: Surely Salieri would have heard of Joseph Haydn, who was equally renowned for his music AND much more successful in life than Mozart. Surely that would make him question if Mozart was truly God's [one] Instrument for his generation.
Salieri loved Mozart's music when he heard it. But then he met Mozart...
Mozart and Haydn held each other in very high regard, and Salieri must have disliked being the third wheel.
Genius Bonus: In the movie, when Mozart calls for different composers to imitate at the keyboard. Mozart's reaction to someone shouting to play it like Gluck ("Bo-ring!") is all the more funny when you know that Salieri was one of C. W. Gluck's most prized pupils.
Emperor Joseph's quip about the silent ballet: "I don't understand. Is it modern?" John Cage's 4'33" was composed in 1952, nearly three decades before Shaffer wrote Amadeus; it was often cited as an extreme example of "modern" music as no instruments are played in it.
Mozart. He's an obnoxious, vulgar nutjob with an annoying laugh, but you still can't help but feel bad for him. It helps that he has somewhat of a childish innocence about him too, and generally means well.
Hell, Salieri might even get points here too. A cold, snide, petty murderous bastard he may be but its clear that he absolutely hates himself and is reminded every day that there is someone better (who doesn't even appreciate their talent).
Tearjerker: The death of Mozart. And then in comes the Lacrymosa...
Only a handful of people attend his funeral, and custom dictates they cannot follow the funeral cortege to the burial site. Which is just as well, as at the cemetery the body is unceremoniously thrown into a mass-grave, with lime dumped on top to aid decomposition. Also serves as a tear-jerker on a meta-level, as until this day no-one really know exactly where Mozart is buried.
There's a local legend that the nightingales sing more sweetly over Mozart's grave than anywhere else in Europe.
The Woobie: Mozart's young maid who's in Salieri's service. She never really knew what was going on, but what happened will haunt her forever.