We study the wavelength dependence of the dust emission as a function of position and environment across the disk of M 33 using Spitzer and Herschel photometric data. M 33 is a Local Group spiral with slightly subsolar metallicity, which makes it an ideal stepping-stone to less regular and lower-metallicity objects such as dwarf galaxies and, probably, young-universe objects. Expressing the emissivity of the dust as a power law, the power-law exponent (β) was estimated from two independent approaches designed to properly treat the degeneracy between β and the dust temperature (T ). Both β and T are higher in the inner than in the outer disk, contrary to reported β − T anti-correlations found in other sources. In the cold + warm dust model, the warm component and the ionized gas (Hα) have a very similar distribution across the galaxy, demonstrating that the model separates the components in an appropriate way. Both cold-and warm-dust column densities are high in star-forming regions and reach their maxima toward the giant star-forming complexes NGC 604 and NGC 595. β declines from close to 2 in the center to about 1.3 in the outer disk. β is positively correlated with star formation and with the molecular gas column, as traced by the Hα and CO emission. The lower dust-emissivity index in the outer parts of M 33 is most likely related to the reduced metallicity (different grain composition) and possibly to a different size distribution. It is not due to the decrease in stellar radiation field or temperature in a simple way because the far-infrared-bright regions in the outer disk also have a low β. Like most spirals, M 33 has a (decreasing) radial gradient in star formation and molecular-to-atomic gas ratio such that the regions bright in Hα or CO tend to trace the inner disk, which makes it difficult to distinguish between their effects on the dust. The assumption of a constant emissivity index β is obviously not appropriate.