Many systems in nature exhibit avalanche dynamics with scale-free features. A general scaling theory has been proposed for critical avalanche profiles in crackling noise, predicting the collapse onto a universal avalanche shape, as well as the scaling behavior of the activity power spectrum as Brown noise. Recently, much attention has been given to the profile of neuronal avalanches, measured in neuronal systems in vitro and in vivo. Although a universal profile was evidenced, confirming the validity of the general scaling theory, the parallel study of the power spectrum scaling under the same conditions was not performed. The puzzling observation is that in the majority of healthy neuronal systems the power spectrum exhibits a behavior close to 1/f, rather than Brown, noise. Here we perform a numerical study of the scaling behavior of the avalanche shape and the power spectrum for a model of integrate and fire neurons with a short-term plasticity parameter able to tune the system to criticality. We confirm that, at criticality, the average avalanche size and the avalanche profile fulfill the general avalanche scaling theory. However, the power spectrum consistently exhibits Brown noise behavior, for both fully excitatory networks and systems with 30% inhibitory networks. Conversely, a behavior closer to 1/f noise is observed in systems slightly off criticality. Results suggest that the power spectrum is a good indicator to determine how close neuronal activity is to criticality.