We characterized the population of galaxy clusters detected with the Sunyaev-Zeldovich (SZ) effect with Planck by measuring the cool-core state of the objects in a well-defined subsample of the Planck SZ catalogue. We used as an indicator the concentration parameter. The fraction of cool-core clusters is 29+/-4 per cent and does not show significant indications of evolution in the redshift range covered by our sample. We compare the distribution of the concentration parameter in the Planck sample with the one of the X-ray selected sample MACS: the distributions are significantly different and the cool-core fraction in MACS is much higher (59+/-5 per cent) than that in Planck. Since X-ray-selected samples are known to be biased towards cool cores due to the presence of their prominent surface brightness peak, we simulated the impact of the 'cool-core bias'. We found that this bias plays a large role in the difference between the fractions of cool cores in the two samples. We examined other selection effects that could in principle bias SZ surveys against cool cores, but we found that their impact is not sufficient to explain the difference between Planck and MACS. The population of X-ray underluminous objects, which are found in SZ surveys but missing in X-ray samples, could possibly contribute to the difference, as we found most of them to be non-cool cores, but this hypothesis deserves further investigation.