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I have heard that Ravi Shankar once said that ragas are not jazz. However, Ravi Shankar had a significant influence on some jazz performers.
How does improvisation in Indian ragas differ significantly from jazz melodic improvisation?
They both seem to improvise from sets of notes in particular patterns.
For much of Jazz, the harmonic changes provide a key forward push. Melodic choices have to work within that framework. In Indian music, there are no harmonic changes to exploit for forward motion or melodic development.
With some "smooth jazz" and some modal jazz, the tunes stay in a single key or mode, and there is more similarity to Indian music, but even there the logic it different in that with Western music, the melodic theme or the rhythm section has a big role in driving things, as well as references to our typical song structures.
With a Raga, there is no theme, per se. A Raga is more like a set of rules as to how you progress though the mode than a melody in its own right. Certain notes are designated for emphasis or avoidance, and for being approached or left in a particular manner. With Jazz, an improvisor is left more free to determine how to proceed.
There is a fine "less is more" quality to a Raga. The constraints and consistency serve to give each note in the set its own personal character or meaning, where in Jazz, even with modal, we are often hearing/thinking about notes as chord tones vs. color tones vs. dissonances, using our familiar key and chord structures as references.
There is undoubtedly a lot of cultural context that I am ignorant about, as well, that add to the differences. Also, the way rhythmic structure is conceived has many differences but I only have a rudimentary knowledge of this. Probably should leave it to someone else to explain.
I'm just responding from the few classes I've taken in Indian music and from what I could make out for myself about what people are doing when they improvise in it.
That said, there are some patterns that turn up in both forms, as well. It's not totally different.
An extension to @JoseDavid's answer, On the subject to rhythm, the structural pattern is more complex in classical music, There are 16 beats(teentaal),7 beats(rupaktaal),12 beats etc (Ektaal) rhythmic cycles in icm, on based of that the raga composition is created and improvised in various tempos i.e Vilambit (very slow), madhya (medium) and drut(fast) and some time ati drut (very fast) also, depends on performer. Also one single classical raga is represented in all tempos with similar or different talas(rhythms) in 3 tempos as mentioned above.
The raga improvisation is done according to below factors, 1.The Notes that are going to be used. In each raga there are prominant notes, resting notes and restricted notes. 2.How the notes are going to be used. This means how the transition from one note to another note is done, jump from one note to another, some ragas are based on meend (i.e gliding on notes), peremutations and combinations of notes. 3.Concept of mircotones. Some ragas are having similar notes in ascending and descending and generates similar kind of melody but still different moods ,they can only be differntiate by microtones. These microtones are called "shrutis", there are 22 shrutis there in icm. also in india classical music not only the fixed notes are used, but also the places in between the notes are also explored.
This should be a comment to Phil Freuhofner's rather good answer, but it's to long for a comment, sorry for that.
First a clarification, properly a "raga" is not a style or genre, it is the collection of notes selected to execute a piece in Indian Classical Music (as some sources prefer to call it, or Carnatic Music, if referring expressly to the more pure Indian tradition from the South India).
The concept of raga has some similitude with those of scales, or modes, in Western Music, but is not the same thing. If I understand correctly it is perhaps more similar conceptually to a dodecaphonic series, but again, with differences.
Adding to Phil's answer, on the subject of rhythm, the structural element is the tala, which is a framework for the rhythmic improvisation and it's a rather more complex and long structure than a meter or a pattern. A piece usually starts with slow tempo and no clearly defined rhythm, transits to a quicker and more pulsating section and finally the percussionists join in as the music gets faster and more rhythmically complex, until it gets to climax.
This article is a good introduction to the theme of Indian music, I think.
Jazz improvisation is generally based over a chord sequence, which is itself usually the form of a popular song (or a song composed specifically as a vehicle for improvisation). It is the form of that song which keeps everything together. Soloists do their thing for some number of "choruses", that is complete repetitions of the theme, which is often 32 bars in length. Rhythmically we are usually in 4/4, tempo can be pretty much anywhere.
In Indian music there is no concept at all of chordal progression. In fact the idea of harmony, in the western sense, does not exist. (Maybe there is some kind of background drone provided by tanpurah or similar.) The structure is dictated by the time signature (taal) and the musicians keep track of the whole rhythmic cycle for some given numbers of bars, often with complex syncopation and cross rhythms. There are differences in how this is done between North and South Indian systems. The choice of notes used is given by the raga, which is more or less independent of the taal I think (but which also implies certain ornamentations and so on).
There is also a "song" element to Indian music, the songs generally being devotional folk songs, but these don't seem to be used as improvisational vehicles.
The similarities really end with the fact that there is improvisation. Raag are a much more formal and structured performance than would generally be found in jazz. A raga is a collection of notes, musical phrases and rules that give an emotion or feeling and are often associated with a time of day or season. A performance generally consists of the Alap, a free form and free rhythm introduction to each of the notes and key phrases in the raga. The performer brings out each note with a range of phrases first descending, then through the middle octave and up to a higher octave before descending. At this point generally a composition is played which provides a theme - once played straight, the performer starts to ‘stretch’ the composition, improvising in different sections.
A number of raga have the same notes but they key phrases and rules are slightly different giving them a different feeling and emotion. The harmony in Indian classical music is static and often provided by a tanpura, a drone based instrument that will play the performers choice or one or two notes from the raag. The notes of the raag are always played then relative to the drone notes which places more emphasis on the intervals between the melody and the harmony that can probably be felt in more complex harmonic music.
There are similarities but really these are quite general - jazz l, even modal jazz, is usually harmonically complex whereas Indian music had much simpler harmony generally made by intervals between two notes (medley and drone) and of course the relative harmony that the audience creates in their mind from the intervals between melody notes.
