An interesting and complex topic, on which many have reflected. Discussing it exhaustively is almost impossible...
However, I would like to focus for a moment on the terms, to make some distinctions that seem right to me.
1) SYNTAX, namely the rule of articulation of harmonic or melodic units, exists in almost all compositional systems. In modality, in counterpoint, and in tonality, there are rules that dictate which chords might sound good after each chord or that, in any case, study the different effects of different successions. Let us remember, however, that almost no syntax is truly normative. That is, there are no obligations that have never been disregarded.
2) The syntax of many compositional systems is TELEOLOGICAL; it identifies, that is, tensions that lead the musical discourse toward a goal. Tensions toward the tonic or toward the finalis are an example of teleology. But there are different ways to approach this teleology. We can use the usual triadic logic of thesis-antithesis-synthesis, typical of sonata form. The "negation," consisting of modulation to the dominant, is nothing more than a "tension" that leads us more strongly to the tonic. But not all teleologies are so clear-cut. The various cadences that precede the cadence to the finalis in a chorale, for example, do not create "tension" in the pure sense; they are not tense toward the finalis. They are merely distractions, from the point of view of teleology...
3) As a consequence of the previous point, one must strive to distinguish between TELEOLOGY and NARRATIVITY. A composition can be narrative even if it has no destination. And vice versa. In the history of harmony, the greatest blow to teleology, and to tonal syntax, is the defunctionalization of tension chords, especially sevenths. That is, a chord that should lead to the tonic, at a certain point goes mad and leads somewhere else. Paradoxically, these defunctionalizations have gifted us some of the most narrative compositions in history, given that life, as we know well, is certainly not a straight road with a beautiful, clear, and gigantic pre-established goal. Excluding death, obviously. If teleology identifies a goal, narrativity should identify and make sense of a path.
4) From the previous point comes another consequence, also outlined by Frank. Narrativity is not only harmonic; it is not made only of tensions and harmonic functions, but is more complex. The role of REPETITION, for example, is fundamental. And it is in a discourse "between" teleology and narrativity that the concept of ELABORATION is situated. Because, let us remember, elaborating a musical element does not serve to reach the goal, but to delay reaching it, to create a path. In fact, paradoxically, if we consider the repetitive factor of elaboration—reformulating—we could say that variation is explicitly a method for forcing the temporal direction of music.
5) This is a point that greatly interests me and is somewhat fashionable. Music is the art of time, it is often said. But composers have always sought to vary the temporal flow, to force it, almost abolishing it or turning it around. Inverted canons, rondo forms, extremely long cycles of variations, ostinato basses, are all musical means that can easily be interpreted as forcings of the natural temporality of a piece. TEMPORALITY should be the temporal dimension suggested by a musical piece, and it is different from TIME, which is chronometric. Playing with syntax, with rhythm (especially harmonic rhythm), and with teleology serves to influence our temporal perceptions, and it is a game that has been played very often by composers, especially in the 1900s.
6) There are compositions that criticize teleology, compositions that criticize narrativity, and compositions that criticize both, or neither. In a general sense, it is always good to remember that there are also dodecaphonic operas (not serial, as far as I know...) and that things like thematicism, elaboration, and variation are widely usable in non-tonal systems.