While the foundations of Western music are consistent in both classical music and jazz, these two genres take music theory in different directions.
There are places where there are strong similarities (remember, this is just talking about music theory - we're not talking about general similarities between jazz and classical music): a Cmajor7 chord (not necessarily the voicing) includes the notes C, E, G, and B, all notes found in a C major scale. While a blues scale (in C, the notes are C, Eb, F, F#, G, and Bb) seems nothing like the C major scale, this is because the C blues scale actually has more in common with the Eb major scale, and the only note in the C blues scale that goes against the Eb major scale is F# (or Gb), which is not intended to be emphasized in a blues scale anyway.
Put simply, many of jazz chords and scales have a lot in common with classical music scales - but, at the same time, they're not quite the same. The differences are somewhat small, but important.
We will begin by looking at the C blues scale - which, remember, is similar to the Eb major scale. Here are the two scales for comparison (the note an octave above the root note is not included):
Eb major scale
Eb, F, G, Ab, Bb, C, D
C blues scale, with Eb as the root note
Eb, F, Gb, G, Bb, C
Notice how both scales share five notes: Eb, F, G, Bb, and C. That's a lot in common. Yet, the two scales, when used to create musical phrases (as in improvisation), sound different in the following way:
If you try using the notes of the Eb major scale to improvise, you'll find that the scale does not lend itself to improvisation. Sure, you get a few phrases that are okay here and there, but mostly not. The C blues scale with the Eb note as the root, on the other hand, does not lend itself to improvisation either, but sounds much more musical than the Eb major scale. Even if you just run up and down each scale, one at a time, you will find that the blues scale sounds more musical than the other one.
What notes make the difference between the two scales mentioned here? Well, the Eb major scale has some "extra notes" that do not sound especially musical when you're in the key of Eb. In this case, those notes are Ab and D; for some reason, while musically they are correct, they simply do not blend well with the Eb and the rest of the major scale. So jazz music (in particular bluesy music) takes those notes out (not consciously, of course), and what you have left are the following notes: Eb, F, G, Bb, and C. Four of those five notes make a chord that jazz musicians are familiar with, the Eb6, which is musically one of the richest chords in jazz.
The removal of the Ab and D notes give the scale more unity than existed in the original Eb major scale. The Gb note, which jazz musicians add to make the blues scale, should not be emphasized, but should just be passed over quickly to add interest.
To put all that simply (at least, fairly simply), the notes changed between the two scales make the jazz scale (the blues scale with the root note changed) less musically complex than the classical scale (the Eb major scale). Less musical complexity means more unity, and also opportunities to add more musical complexity to a chord sequence. This applies to both the chords themselves and the improvisation built upon those chords, since the two are closely connected - chords are clear statements of a certain tonality, and improvisation involves a collection of notes that fit and often suggest that tonality (sometimes, more clearly than at other times, depending on the improviser and the chord progression).
All of this has a lot to do with why jazz musicians (with the exception of those in traditional jazz) generally avoid chords like "C major" or "F minor" with no sevenths or sixths included - "plain" major or minor chords don't provide enough information to the improviser about what the chord is, and therefore improvisation based completely upon these chords - rather than any other source, such as the melody - will probably not be interesting. Think about it like this: it's a graph. On one end of the graph are chords that are too simple for most improvisation, and on the other end of the graph are chords that are too complex for listenable improvisation. Unfortunately, modern jazz improvisers like the complex chords, making the music a challenging listen for anyone but the extremely musical. The best is the middle of the graph: the region with chords like sevenths, and sixths - rather than ninths, elevenths, and thirteenths - and chords in the range of dominant sevenths, minor sevenths, and major sixths. With these chords, interesting musical phrasing can be developed that is neither too simple nor too complex.