"Fundamental differences" is something that is open to interpretation. Jazz, a great-great-great-great step-grandchild of classical music certainly shares some of the rich musical heritage of European and African traditions which gives it "hybrid vigor."
Nevertheless, classical music improvisation delves into many areas that jazz only hints at.
Here's Robert Levin lecturing on "Improvising Mozart"
My answer here is a very rough summary of the book "Improvising" by Gerre Hancock which was strongly recommended to me by church organists and ballet school pianists back in the 90s when I first started investigating melody-based improvisation. Sorry jazzheads, traditional jazz is mostly based on improvisation from the chord substitutions of the original melody, a fact confirmed by a highly recommended book, "The Jazz Theory Book" by Mark Levine. And I grew up on and love jazz. But I'm a bit weary of the average jazz trio playing melody once through, then noodling on the chord changes for a while and then returning to playing the melody once before ending. The striking exception to this gross over-simplification is Bill Frisell, who leans very heavily on melody-based improvisation.
Here is a direction to pursue in classical improvisation:
The Scale (each of these exercises are in 8 bars/measures)
- Create a melody diatonically, using only one octave and starting and stopping on the tonic.
- Create a melody as before, but add another voice with the melody as either the top or bottom voice.
- Create a scale melody as before but add a third voice
- Add a fourth voice and move in a circle of fifths, now moving out of the 8 bar format into varied formats.
- Avoid all accidentals, playing only on the white keys, until you have mastered this. Then start introducing chromatics.
- Create phrases by improvising answers to questions. Call and response is an incredibly strong musical trope that you can hear in all genres in music. It evokes something very primal within us.
- Create answers in two, three and four voices
- Start diatonically and then introduce chromatics.
- Improvise an interlude of known length, with or without a change of key
- Using the 8 bar phrases, create a song of the form AB where A is a 4 bar question plus a 4 bar answer and B is a 4 bar question and 4 bar answer in a different key which returns to the key of A.
- A bit involved to go into here, but the sonata form is:
- Exposition: Introduction of question and answer.
- Development: Question and answer are stated then combine.
- Recapitulation: Question leads to answer then a triumphal ending.
- Yes, we're still improvising here. Improvising a sonata is part of the exam that French organists have to take. They are given two themes and then have to improvise a sonata. Robert Levin, a noted interpreter of Mozart, does this in concerts.
A simple canon is when the lead voice makes a statement, then pauses while the following voice repeats the statement and makes a new statement and around and around.
Improvising a fugue and counterpoint is unfortunately beyond my competence to explain well. I have composed and played fugues but they are mostly of fascination to me and not so much for the audiences I play for.
So you see that modern rock and jazz have yet to dive deeply into truly rich sources of improvisation. I await this moment with hope and joy.