OP, What you call modern harmony isn't the same what theorists call "modern harmony", which is harmony from aesthetic trends and styles of circa 1880 - 1930, and fairly complex to explain with roman numeral analysis. Just think in music by Mahler, R. Strauss, Debussy, early Schoenberg and Stravinsky, Bartok, Hindemith, Milhaud, Honegger, Vaughan Williams, Villa Lobos et cetera.
After this distinction being made, let me call pop harmony instead.
Pop harmony is a very simplified reduction of late Romantic, golden age Hollywood, 80's rock, 70's dance music, 60's jazz and 50's country and blues harmonic practices and clichés (think of the lick or 4 chords songs archetypes). All of this mixed up and simplified to communicate the easier and faster way with listeners. Now, in 2020, it's even more simplified - as most pop singers cannot sing at all today without heavy use of autotune and other post-production tricks, so is very very common to hear songs with 3, 4 or 5 different chords while the main melody is just a rhythmic repetition of the supertonic from music's key scale most of the time, as supertonic tends to fit well in virtually every harmonic function. Kind of McMusic menu.
Classical styles, for other hand, are built on heavy usage of counterpoint (even in homophonic settings) and ingenious craft of form. Form, harmony, melody and cadences are closely tied together - sometimes it's so tied that one cannot separate these entities in analysis. So, every little detail is obsessively designed and crafted - and meaningful in context.
The main difference lies in there.