First, remember that "classical harmony" is an attempt to study and understand what composers did over nearly a 300 year period. Walter Piston referred to it as the "Era of Common Practice." Principals of harmony as presented in this study are not rules.
This common practice period covers music as diverse as Baroque composing methods with its usually quick and regular harmonic rhythm that underpins contrapuntal techniques, rhythmic pace, and its spinning out a single idea through a movement, to Classical methods that often has a slower and often assymetric harmonic rhythm that underlines primarily homophonic techniques (counterpoint was originally avoided), and incorporating several ideas, both complementary and contrasting, to Romantic period usages that among other things expanded the functional harmonic pallet as well as incorporating non-functional harmonies for color. While Baroque usages of harmony and other elements my be perfectly normal in Romantic Period music, Romantic harmonic usages would be out of place in the Baroque. Unfortunately, the textbooks and teaching of harmony obscure this diversity and helps to cause lots of confusion.
As far as any "breaks" in this supposed continuity of harmonic practice, one could answer a.) that it started 2 days after the concepts were resonably in place, b.) the break-down began (or at least becomes apparent) in much of Wagner with his constant deceptive changes in direction that obscures any tonal center, c.) Late Liszt,some which foreshadows techniques used by Debussy, c.) Schoenberg and his exploration into atonality. (Note: Schoenberg's explorations in atonal chromaticism was essentially a continuation of the breakdown already forshadowed by Wagner.)
What breaks did occur with "common practice harmony" originally was primarily limited to the art music of the early 20th century and in no way represented a supplanting of one by the other. The popular music (including early jazz and blues) was fully rooted in this "common practice" harmonic tradition.
As Bob said, jazz and blues are actually a blending of African and European musical ideas. That is why we can claim it as "American" music. If we look at the basic blues harmonic progression, it is simply a I IV I V I pattern (I (x4meas)|IV (x2) I (x2)|V (x2) I(x2)||). How much more European can you get? As far as the "blue notes" (b7 and b3 are most common), these occured melodically over this basic European progression. They did not begin as harmonic elements. Over time, much like the dominant chord with the minor 7th was ultimately accepted as an independant harmony during the Baroque, the blue notes were incorporated into the harmony of the blues, creating the 7th chords of blues in which the 7th is non-functioning but rather a color tone. These harmonies are more closely allied with mixolydian or dorian modes rather than major/minor. (Mixolydian, with its b7 scale step is fairly common in folk music, even that of European descent.)
In the end, much of jazz harmonic practice simply represents a natural evolution of the harmonic princpals in place, just as Romantic harmonic principals are an evolution of that of Baroque and Classical practice. It borrowed heavily from this and impressionist ideas as well as other ideas.
The main difference between 20th century harmonic practice and the "common era" practice is that while the earlier practice was in general usage in the majority of music of the time, it is no longer common practice in the 20th century. Besides this continuance of common practice harmony, there have been numerous other branches that may or may not have its roots in the earlier harmonic practice -- serialism, non-serial atonality/chromaticism, non-functional diatonicism, and whatever else.
Granted, some jazz started to break away from this concept of tonality and explored modal harmonies and other 20th century techniques.
I hope this makes some sense. It is difficult to write about this idea and keep it concise.
Scott
Topic: harmonic function