We study a sample of 61 submillimetre galaxies (SMGs) selected from ground-based surveys, with known spectroscopic redshifts and observed with Herschel as part of the PACS Evolutionary Probe (PEP) and the Herschel Multi-tiered Extragalactic Survey (HerMES) key programmes. We use the broad far-infrared wavelength coverage (100-600um) provided by the combination of PACS and SPIRE observations. Using a power-law temperature distribution model to derive infrared luminosities and dust temperatures, we measure a dust emissivity spectral index for SMGs of beta=2.0+/-0.2. Our results unveil the diversity of the SMG population. Some SMGs exhibit extreme infrared luminosities of ~10^13 Lsun and relatively warm dust components, while others are fainter (~10^12 Lsun) and are biased towards cold dust temperatures. The extreme infrared luminosities of some SMGs (LIR>10^12.7 Lsun, 26/61 systems) imply SFRs of >500Msun yr^-1. Such high SFRs are difficult to reconcile with a secular mode of star formation, and may instead correspond to a merger-driven stage in the evolution of these galaxies. Another observational argument in favour of this scenario is the presence of dust temperatures warmer than that of SMGs of lower luminosities (~40K as opposed to ~25K), consistent with observations of local ULIRGs triggered by major mergers and with results from hydrodynamic simulations of major mergers combined with radiative transfer calculations. Luminous SMGs are also offset from normal star-forming galaxies in the stellar mass-SFR plane, suggesting that they are undergoing starburst events with short duty cycles, compatible with the major merger scenario. On the other hand, a significant fraction of the low infrared luminosity SMGs have cold dust temperatures, are located close to the main sequence of star formation, and thus might be evolving through a secular mode of star formation. [abridged]
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