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The question concerns western classical music. I have tried reading about and understanding the difference between variations and development, and what I understand is that development is more complex: a variation is something which uses a particular motif and alters it, whereas development can involve motifs from various parts of the composition.

I still don't feel very comfortable with my understanding, and it would be great if someone can explain this a bit more clearly.

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To be simple one can say:

Variation: one musical theme/melodic material that already has been presented is the object of the variations. Variations of that material (one already existing material/melody/theme). Variations normally don't change the melody content neither interrupt/cut pieces out of the original material, but can be very creative with all other aspects like rhythm, dynamic, note length and ornaments.

Development: works the material already presented also, but here the melodic material can be developed/altered and/or presented just fragments of the original material; it is also possible the development will include material from other melodies/themes that have already been presented.

It is posible that a variation is very complex, with tonality changes, ornamentations or other changes to the melodie aspect, so it looks like a development, but still only one melody/element at a time.
It is also possible a development to keep the ideas of the theme melody, so it can look like a variation. Still the idea of variations is to use one theme/melody presented and keep the melodic material quite "original", the development going "forward" in music making a sum/resume of one or many ideas presented and/or even with new melodic/harmonic changed to that material.

There is a line of composition called "Developing variation" which makes variations using serial and mathematic models to make variations of a theme, this produces music that is sometimes difficult to associate with the theme and thus is a "developing variation".

Found this related in wikipedia:

Developing variation

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The Harvard Dictionary of Music defines variation as, "A technique of modifying a given musical idea". It is commonly associated with the form of a piece or movement. These variations consist of alterations to the idea such as elaborations, key/modality change, voice exchange, reharmonization and more.

The Harvard Dictionary of Music defines development as, "Structural alteration of musical material". In its definition it gives this nice little distinction, "Development may also usefully be distinguished from variation, the former involving a true structural transformation, the latter merely and ornamental change such as a melodic elaboration or a shift in dynamics or orchestration".

In the Classical period Sonata form was very popular which includes a full section of development.

Development tends to take things and chop them up into little pieces and put them in different places or change the size of them or shape much more than variation. A variation might include some of those aspects but it will retain its structure.

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Though the term "theme and variations" implies that what is fundamentally being varied is the theme, what is usually preserved in a variation is the length, phrasing, and chord progressions that accompanied the original presentation of the theme. So if you listen to something like the first movement of Mozart's K331 --

-- you will hear that every variation preserves the chord progressions (even in the minor mode variation) and length of each chord.

In a development section, the succession of chords is one of the first things that tends to be changed -- the basic motives -- the smaller parts that make up a theme -- may be preserved, but they are often recombined in such a way that the original chord progressions and phrasing make little sense.

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