Music, prior to public education, was a fairly aristocratic study. With a lot of things in music, you also have the subjectivity of taste. A chord progression you think sounds cool might sound horrific to Mozart, but Holst would like it. This has to do with shifts in attitude by a relatively small music community as well as a need to do something "different" from the people who came before you.

Mozart, for instance, is very "clean." If I do a stem and slur analysis of Mozart or even just listen to something he wrote, it will seem like each chord is the natural consequence of the next. It will seem like "oh, there is no other way for this to happen, it was totally natural to use a cadential 6/4 resolution." Mozart seems this way because everything he writes is holistic. He follows basically every rule of contrapuntal technique and combines it seamlessly with the chords to make nothing stand out. From a compositional standpoint, this is incredible, and very hard.

Everyone since Mozart has essentially learned the rules, and then broken a couple in order to innovate. For instance, it's "normal" to use the melodic minor scale by certain conventions when writing four part harmonies. This is to ensure that your Dominant chords pull to the tonic in the way most classical music needs to. Dvorak, famously, didn't do this in his New World Symphony. You see an non-raised seventh during the theme of the fourth movement of this Opus. This makes it sound "Folky," according to my Theory professor.

Different Eras of music history since Mozart are essentially different flavors of rule breaking. Even within that, various composers of the same era will make different choices when they write themselves into corners. If you're composing something by Baroque rules, you will find yourself in situations where you have to choose from one of several bad options. Which options you choose, or how you can innovate and work around this reality, will define you as a composer.

Jazz, pop, electro, and other modern music might sound good to us, but the rules of classical music originally existed because to a trained ear, doing predominant chords "backwards" sounded unsatisfying. But some people might prefer it, and this is essentially why the song Sweet Home Alabama is popular. People prefer different stuff. Globalization, as well as the advent of digital recording, listening, and editing tools, have led to music becoming less purist over time.

You asked specifically about Mozart through the 20th century, but that leaves out a lot of what people often lump in under "classical" music. In your question you even mention the baroque composer Vivaldi, who died 15 years before Mozart was born. And I would argue that beginning with the baroque era (as many conservatory and general music programs do) is also pretty arbitrary, as it refuses to acknowledge the origins of Western notation and harmony.

I have tried to type up an answer to your question, but even with its time period restriction, it's so broad that I'm having a hard time. But here are some things that are often taught for categorizing, in the basic chronological order (please know that I am leaving out a LOT and not everyone would agree that these are the "important" points/people):

Mozart, Haydn, Süssmayr, Salieri: clean, "perfect," often perky, formal, beautiful, rhythmically and harmonically more predictable than the ones below, more sparse than much of what's below

Beethoven: often cited as the bridge from Classical to Romantic -starting to see more sweeping phrases, "emotional," less formally predictable

Schubert, Schumann, Mendelssohn, Brahms: more dynamic and tempo variation than Classical/galant music within each phrase, less dramatic Romanticism than the next group, but still solidly Romantic

Wagner, Strauss, Chopin, Lizst, Rachmaninoff, Mahler, Puccini: height of Romanticism/late Romanticism, getting more and more "emotional," "emo," heavier ("more lush") orchestration, louder, grander, and bigger

the 20th century gets interesting in that trends become more and more eclectic and divergent. You have the lyricism and yet dissonance of Les Six, the atonal music of Schönberg, the Neoclassicism AND dissonance of Stravinsky, electronic music of Wendy Carlos, a new supposedly uniquely "American" style with Aaron Copland, minimalism of Steve Reich and Philip Glass, composers doing both opera and movie scores, indeterminacy of John Cage, and much more. It's hard to put all these into the same category in terms of style, so when we speak broadly, we just put them into a time period.

Now that we're in the 21st century, this trend of divergent but coinciding styles continues, and the most successful art music composers today may write anything from romantic-ish neoclassical-ish sort of Renaissance-inspired music (Eric Whitacre, Morten Lauridsen) to pieces that consist of grunting into a feedback machine, or have long sections of speaking or improvisation, or include stage directions, or use homemade instruments. Often, our contemporary music is notated with original and inventive methods. New music is a fascinating field, and premieres are happening all the time.