When I first started working with just intonation in the mid-seventies, it was cool but definitely not yet the minor fashion it has become; my work in the area was even actively discouraged by many musicians, composition teachers among them. Now that tuning to intervals of small and smallish whole number ratios has become substantially more acceptable — and alternative/microtonal tunings in general, including historical tunings and tunings outside the western traditions — the rationale for going to the extra effort required to get intonation right is often obscured, sometimes even by extravagant claims by proponents. Adorno famously "defended Bach from his devotees"; let me try a similar defense for just intonation.
Often, just intonation is promoted as more "natural" or better representative of the natural tendencies of real performing music. This argument doesn't work for a number of reasons. First, because we have little or no data to confirm that just intonation is the prevailing habit or intention. Intonational control is hard, and musicians in all traditions, even the most esteemed, show wildly varying skills in pitch control and replication. There are special instances, in barbershop singing, for example, in which practice clearly includes targeting harmonies to just intonation, but this done intuitively rather than formally, and the melodic practice required to yield those harmonies is complex, and species of temperament, informal and intuitive in its own terms. Further, the "natural" intonational tendencies of musicians are far from universal, with the particular instrument and repertoire and practice tradition playing large roles, as well as the unique intonational preferences of individual musicians (the Javanese would call this embat, literally "bouncing", as in the sense in which each person walks with their own lilt). The tendency of western string players towards large major thirds and leading tones is an afffinity with pythagorean intonation, but one which contrasts starkly with the affinity of horn players, for example, to tune to harmonic series segments, but these are only tendencies and actually practices is rather more pragmatic. Instrumentation is also important because not all instruments have simple harmonic spectra, thus, for these instruments, just intonation is not inevitably optimal in terms of reduced spectral beating.
Often, just intonation is promoted in the abstract, by an appeal to reason, indeed to the world of platonic ideals. Your own mileage will necessarily vary on this one, but my own interest is in real, existing music in the world I happen to live in. Nevertheless, I personally find that an abstract approach to just intonation, classifying intervals by the odd or prime factors in their ratios, provides a superb method of sorting through and mapping (typically onto a lattice or manifold) tonal materials and one that, to a large extent, corresponds closely to my experience of those materials in the acoustic reality of music made well.
One argument that does work, and bridges the abstract and real qualities of intervals just intonation is this: Just intonation makes intervals and the tonal relationships among them more explicit. This may well come at the loss of some useful possibilities for tonal ambiguity, particularly those offered by temperaments, but there are alternative forms of ambiguity offered by an extended just intonation and it is a legitimate musical aesthetic to not use such ambiguities.
But, in the end, the most important justification is simply this: with voices and with instruments with simple harmonic spectra, just intonation provides the set of intervals which will maximize sensory consonance. Not taking advantage of this as a composition resource — and, implicitly, the entire range of sensory consonance and dissonance afforded by subtle and not-so-subtle deviations from these intervals — is rejecting one possibility for the qualitiative enrichment of music and, from this point of view, is making music less rather than as much as it might be.
See also this earlier post, What alternative tunings can do for you