Molecular mapping is a supervised method exploiting the spectral diversity of integral field spectrographs to detect and characterise resolved exoplanets blurred into the stellar halo. We present an update to the method, aimed at removing the stellar halo and the nuisance of telluric features in the datacubes and accessing a continuum-subtracted spectra of the planets at R ~ 4000. We derived the planet atmosphere properties from a direct analysis of the planet telluric-corrected absorption spectrum. We applied our methods to the SINFONI observation of the planet β Pictoris b. We recovered the CO and H2O detections in the atmosphere of β Pic b by using molecular mapping. We further determined some basic properties of its atmosphere, with Teq=1748−4+3 K, sub-solar [Fe/H]=− 0.235−0.013+0.015 dex, and solar C/O=0.551 ±0.002. These results are in contrast to values measured for the same exoplanet with other infrared instruments. We confirmed a low projected equatorial velocity of 25−6+5 km s−1. We were also able to measure, for the first time and with a medium-resolution spectrograph, the radial velocity of β Pic b relative to the central star at MJD=56910.38 with a km s−1 precision of −11.3±1.1 km s−1. This result is compatible with the ephemerides, based on the current knowledge of the β Pic system.