  [Abridged] Context. The CoRoT space mission has been searching for transiting
planets since the end of December 2006. Aims. We aim to investigate the
capability of CoRoT to detect small-size transiting planets in short-period
orbits, and to compare the number of CoRoT planets with 2 \leq R_p \leq 4
Rearth with the occurrence rate of small-size planets provided by the
distribution of Kepler planetary candidates (Howard et al. 2012). Methods. We
performed a test that simulates transits of super-Earths and Neptunes in real
CoRoT light curves and searches for them blindly by using the LAM transit
detection pipeline. Results. The CoRoT detection rate of planets with radius
between 2 and 4 Rearth and orbital period P \leq 20 days is 59% (31%) around
stars brighter than r'=14.0 (15.5). By properly taking the CoRoT detection rate
for Neptune-size planets and the transit probability into account, we found
that according to the Kepler planet occurrence rate, CoRoT should have
discovered 12 \pm 2 Neptunes orbiting G and K dwarfs with P \leq 17 days in six
observational runs. This estimate must be compared with the validated Neptune
CoRoT-24b and five CoRoT planetary candidates in the considered range of
planetary radii. We thus found a disagreement with expectations from Kepler at
3 \sigma or 5 \sigma, assuming a blend fraction of 0% (six Neptunes) and 100%
(one Neptune) for these candidates. Conclusions. This underabundance of CoRoT
Neptunes with respect to Kepler may be due to several reasons. Regardless of
the origin of the disagreement, which needs to be investigated in more detail,
the noticeable deficiency of CoRoT Neptunes at short orbital periods seems to
indirectly support the general trend found in Kepler data, i.e. that the
frequency of small-size planets increases with increasing orbital periods and
decreasing planet radii.
